Some of the advice here may depend on the level you play on. On Emperor and below, you can basically horse rush yourself into a very dominant situation with about any civ.
On Immortal, the AI's develop quicker and have large armies and horses and nice middle aged units like cho-ko-nus and longbows start to become obsolete much quicker.
Anyhow... and noting that I haven't played as Greece yet, I find Mongols, China, and Songhai to be nice warmongering civs.
Mongols with a fair supply of horses can easily wipe out a land mass with several AI civs on Emperor and below and without ever stopping attacking. Just keep a couple horsemen for taking cities and busting up the occasional cannon you may see, along with a keshik army and a Kahn and this is easy. On higher levels, you may not be able to sweep the continent clean in one go, but you should be able to cripple a nearby AI or two and expand via conquest into a very competitive later game situation.
The extra movement of a Kahn is extremely useful. In my last Immortal game, My Kahn was able to move along the roads on the eastern front and support what ever attack I was making during the part of the turn and then move again, to another part of the front to support that attack and then still get to safety to aid my defence vs Alex's counter attack. This went on for at least 100 turns and with industrial/modern era units (noting that I had an extremely horse poor start and couldn't do an early rush from my penisula). At game end, the Kahn was still useful supporting my mostly GDR army that pushed thru to Athens.
China is a good civ for warmongers also for three reasons. 1) Better GG's and faster generation of them. 2) Their nice UU, the CKN, although as usual on Immortal and above the dominant time frame for this UU is considerably less than on Emperor and below. 3) Their superb UB, the Paper Maker which is a Library with 3 free GPT per building. An army consisting of a few CKN's, a couple horsemen, and a GG is very effective vs equivilent age AI units and can take cities quite nicely. The economy, aided by the great UB, can also afford to purchase more units as needed and also to upgrade them when that time comes, and can usually rush buy the colosseum or circus if critical happiness is an issue due to quick conquest.
I find that Songhai is also a very good civ for warmongering. The triple gold from conquering a city is very helpful and results in nice purchases and getting 75 gold rather than 25 from whacking barbs is nice early on, often leading to a quicker Maritime CS alliance. Their UU, the Mandakalu Cav, can in groups of two or three overrun city after city in its time frame (of course a smaller timeframe on Immortal/Deity). The UB is essentially 5 free culture per building and this helps with rapid advancement thru useful SP's.
Certainly with Songhai, I always want to start with Honor for the barb bonus and then the 15% from adjacent unit combat bonus. I'd probably start with Honor for any major early war mongering plan, although I find Meritocracy a near must to keep happy later on.
.. neilkaz ..