Yeah... good luck. There aren't many good games set in Ancient Greece and China. It seems that games set in the west will be Roman-era, and in the east it will be 3 kingdoms (any game that says "warring states" is probably referring to Japan). Or, you know, they'll just make another WW2 game.
I remember Tin Soldiers: Alexander the Great being a lot of fun, although it's been almost a decade since I last played it.
Hegemony Gold has really repetitive gameplay, but it has two campaigns which follow the Peloponnesian War and Rise of Macedon very closely. I enjoyed it at the time.
Alea Jacta Est is an Ageod game, so if you like Ageod then check it out. The base game is obviously set post-Alexander, but it gets honourable mention because it has a campaign about a King Pyrrhus invading Italy.
Speaking of Pyrrhus and Hegemony, Hegemony III has an expansion which is based on that same period. I haven't played it, need a new pc I think.
Slitherine made a game called Sparta which is kind of a 4x in ancient greece. It's a bit shit, really. They made a similar game set in an earlier period, too.
Slitherine's Fields of Glory was fun back in the day. There was a decent sized online community (single player was total shit), and it is a very fast paced and light battle simulator. No campaign mode.
HPS gaming made a bunch of titles set in the period. I hate them, mostly because of the god awful interface and graphics.
On GOG there's the Great Battles collection. It's p good, go get it if you like the look of it. Only $6 now, and I'm sure they'll eventually have a sale where its even cheaper.
There's a game in early access on steam right now set in Ancient China. It's a 4x, it shows some promise but I wouldn't say it's good yet. If you really like civilization then go get it.
Take a peek at Apotheon, too. It's a platformer, but if you like ancient greece you might be interested. Curse of Issyos is another platformer.
If we're going down the mythology road, then Populous II has a strong greek theme. It's also the best Populous title, and an awesome game in general. The city-builder Zeus and Poseidon is good, too.