Gonna be structuring this post as premise, followed by last turn overview, followed by achievements and some notes on how to achieve them. If you don't have much time and want some tangible info, scroll down and prioritise the achievement notes.

So first off, I was playing a singleplayer large world, normal pace and metropolis difficulty against these AI:

My goal going into it was "Edgar always seems to ascend first and pick Harappans, I want to see if it's actually possible to play as them". Then I actually succeeded, and it became "if Harappan is so special, can they win with one city and my questionable skill at this game?". It went pretty damn well

The continent I started on, featuring Teal and Brown. Teal was sort of a douche in the early game, making me lose my chance to get the achievement for producing no new units; I erased their second city (to replace with an outpost) and vassalized what remained, freeing them about halfway through the industrial era when I realised I probably wouldn't be unlocking any more vassal interactions, couldn't do anything with them and wanted more people to import luxury goods from.

Brown, on the other hand, was quite amenable from the get go after I smoothed over a few crises by renouncing the border friction they caused. They went on to become an ally from the early classical period onward.

There were a few regions in the game that I never settled in time, like that awkward enclave with Brown, because all that territory attaching cost a HELL of a lot of influence. It was awkward to be saving influence for attachments that would benefit me AND cockblock other players by claiming all the resource-rich/strategic regions. The last region I attached in this run cost over 10,000. I took the civic that allows you to buy outposts with money, but then cancelled it because I thought I'd be able to use money or influence. "Allows to claim, attach and merge territories with money instead of influence" is a bit misleading, the "allow" can imply it's a new option rather than a replacement.

above is the second continent where Orange (Ed, the conniving son of a gun), Blue, Green and Pink spawned. Ed, of course, has some incredible advantage from existing, and kept bullying the others until becoming the last domino to ally me during the contemporary era in my quest for total world peace. I just feel blessed he didn't spawn on the same continent and steamroll me, for once.

Here's the new world. It was settled while Ed was trying to conquer his neighbours, so our alliance bloc had a pretty peaceful go of things. The independent cities were a bit of a hassle at the start but my friends very quickly assimilated them. You can see I had a few outposts I never managed to attach... I probably could have, if I didn't claim any cultural wonders.

Across this game, the second continent guys other than Ed were pretty chill for the most part. My general dominance kept them from being too belligerent, and then it was a case of just renouncing grievances and buying luxuries until we built up enough trust to ally, one by one.

It probably helped that my religion suffused the entire planet other than that spread by Independent factions.

It also didn't hurt that my culture dominated the starting continent, including both my neighbours' capitals. There was a period during industrial where I kept getting osmosis events from Brown (then France) even though my culture was dominant, which I'm not complaining about (WOOOOOOO free science!) but do wish the UI would explain better.

These technologies are as far as I got before the end. I believe Brown or Orange may have had some techs I didn't, but I had the most overall.

Syndey Opera House and the Statue of Liberty were in the bay south of my capital, and you can see Angkor Wat and Halicarnassus between the two.

In the capital area itself, you can see Stonehenge, Empire State Building, and Topkapi Palace along with Angkor and Halicarnassus. My capital area isn't as sprawly as I expected from a one city run. It's mostly just farms stretching east with a couple makers quarters (industry). There were some more farms, makers quarters and Harappan Canals dotted around each attached territory. I never actually built a science district until maybe turn 285, there were always other things to make like wonders or misc districts or extractors. Even then, I only built science districts so I could clear the related infrastructure off the available buildings.

Here's the result screen, actual proof I finally beat Ed. I won't bother posting statistics graphs unless asked, but I think it's worth noting that my Fame remained highest throughout the game after turn 20, my capital never went below 100 stability after turn ~85, my population remained highest from start to Industrial era (where Pink started to skyrocket), and my religion eclipsed everyone until a point on the graph where it instantly drops, which I think corresponds to the atheism/secularism civic choice. Oh, and for the final 2 or 3 eras I consistently had more population than available jobs, because districts were the lowest priority (which probably also explains the stability, along with all the wonders).

Now, for the bloated loot bag of achievements:

Everybody's friend - be allied with 3 or more players simultaneously

was just a matter of time, as I kept renouncing grievances and buying luxuries and being influential.

Extremist - be at maximum or minimum values on all ideology axes simultaneously

I just sorta followed what choices I want, which got me most of the way there. I got 3 axes to extreme through capital event choices, and nudged the last axis by looking through civics I'd previously ignored to save influence until I found one that worked.

Pacesetter - be the first to reach all six eras in a single game

This one I'm going to attribute to being Harappa. I don't think you can overestimate the value of having every worker slot filled ASAP. Probably would've been more ridiculous, had I bothered with different districts.

Unplugged - win a game with the AI difficulty set to Metropolis

I think I would've got here eventually with some practice, Harappans or not.

Disconnected/Terminated - win a game with the AI set to Nation/Humankind

These achievements are bugged. It's evident that some sort of bug/scripting error/oversight includes all the difficulty achievements above, rather than below, the one played on.

Captain of Industry, Money for nothing/Money money money, Eureka!/Nerdopolis, Just eat  - produce at least 1000 industry per turn in a city, produce at least 500/1000 gold per turn in a city, produce at least 500/1000 science per turn in a city, produce at least 500 food per turn in a city

Exploiting a lot of tiles with districts that can use that particular resource, and keeping up on infrastructure. I'd place the emphasis on infrastructure, given that I didn't have any science districts until the end when I built maybe 5 or 6 of them. Another thing that makes up for the science is you can get a treaty with allies that boosts it by 5% for both parties. 5 allies means +25%, unless the bonus only counts the first time, which I'd hope isn't how it works. The last thing I believe really swung these in my favour was importing tons of luxuries, since stacking them stacks the buffs they give to flat or % increase in industry, science and other things, and you can get infrastructure that gives you extra gold for each trade route active.

Key influencer - all empire capitals are in your sphere of influence

I'm not actually sure how this one works. Influence seems directly related to culture spread, but my culture didn't dominate the world. My best guess is "sphere of influence" means exerting cultural pressure on, regardless of whether it's progressing in your favour, meaning my high influence output and trade routes were probably doing it... so that's what I'd focus on for this one.

Talk to the hand - finish a game where you refused every received demand

This one was pretty simple, just say no. It may be an issue if you have a slow start. I said no to Blue, and they tried to take my border territories... didn't go so well. Have the economy/army to back yourself up and it should be a breeze, you can probably achieve it without trying during a regular "good" game.

Seven wonders of my world - construct seven cultural wonders in a single game

Industry and influence. Especially influence in later eras, when claiming a wonder to build can cost 7000 or 8000. You'll also want to avoid delaying eras, because AI (especially Ed) will snap them up as soon as they can afford the influence.

Spartan - win a game with a single city

One of two main achievements I was going for this run. I'm still not entirely sure whether the criteria are a) never fully possess more than one city for a full turn or b) only have one city by the end of the game. I occupied two of Blue's cities during their war against me, but I razed the admin centre of one before the war ended and then vassalized them. I also assimilated an independent city at one point, but started razing it on the same turn and the raze finished at the start of the next turn. I'd be curious to see if anyone achieves this by absorbing all other cities before the end.

Rave culture/An endless civilization - Win a game while keeping a culture for 4 eras/the same culture through every era

The other main achievement I was going for this run. I could continue with my Harappan agenda and say they were the pinnacle of why this worked, but based on the lessons I've learned from this run, I'd wager that Babylon, Egypt and Olmec have similar staying power with their legacy/unique building. Still, population is nothing to be sniffed at. I believe there's a strong causal link in the correlation between my population and the other metrics in the other game-end statistics.

The meek shall inherit the earth - Win a game without declaring any wars

The only thing you're prevented from doing is pressing the Surprise War/Declare War buttons. I learned first hand that you can goad another player into declaring on you by refusing all their demands. I almost broke this achievement by declaring on Ed while he was going full imperialist, but decided to just hold my breath. It was rather annoying how no allies called me in to help defend, and there was no option to gift them my units after moving them into allied territory. I suppose they expect me to independently declare on him? If that's the case, it would seem alliances don't actually serve a diplomatic function, as they don't incentivise/deter players when choosing to declare war. At least I get map vision, free science/influence and cheaper imports.

Flawless - win a game against an expert AI persona

So, that explains a lot. Edgar Allen Poe is so stupidly fast at leaving Neolithic (and the picking Harappan>Maya) because he's an expert. I hope he's been humbled by this experience. My best advice for Expert AI is only play with one in your roster, and pray you don't spawn on the same continent. They are so incredibly aggressive.

Behold! - achieve twice the fame of a surviving player at the game end

Should be relatively easy if you steamroll the early game. Just be sure to vassalize rather than fully annex an AI, maybe ally a weak one if you find them in a dire situation. If you're doing well, it's easier for them to be erased than for you to get less than double their fame. I also don't know if it's related to vassalage or bad economy, but I noticed Blue took absolutely forever to reach each new era.

They know not what they ask - win a game without accepting a civics backlash

I'm not sure what a civics backlash is - but I avoided one, so yay!

Life of luxury - possess 5 wondrous luxury resources

I just sort of... kept importing whenever new luxuries became available with other players, and extracting every territory I could claim, and building the mega exploiters when they became available. Didn't even notice when the luxuries became "wondrous". I did notice that imports don't seem to account for the buff when hovering over their icons in the UI, but I got this achievement, so I suppose they must? Maybe it's less of an impact than directly owning them? I'm not sure.

The land of smiles - Win a game without empire stability falling into unrest

I was building outposts/attaching territories whenever I could. I attribute the ludicrously high stability to building a wonder during each era, having a couple garrisons dotted about, and picking one or two stability tenets in my religion.

That concludes this post, I may have to do it in multiple parts if the site throws a hissy fit, I saved on the final turn so feel free to ask for more info I can dig up if you're curious.