I just completed a game with a 300+ pop city. Actually was 600+. I did it by spamming spearmen, which only cost 1gp/turn each. I started this when all my cities were reaching population caps. I just almost continuously pumped out the spearmen, but also threw in infrastructure and some districts from time to time as well.  I had already subdued or frightened everyone else, so I was pretty safe. My capital city was knee deep in spearmen and I just disbanded them all when I was 5 turns from the endgame. Oh, and I also upgraded some of those spearmen to line infantry near the end since they're worth 4 pop instead of just 1. I hope this helps.

Also, I'm not sure if using mods would negate getting the achievements, so I didn't use any.