There are some things you have to keep in mind to win as Rome:
1. Micromanage the start: Good moves in the first turns will pay off, like in every Civ game. That includes to use the legions to make roads and to build a workforce of around 20 Workers in the first turns. (Minor hint: Rome and Neapolis can, given certain terrain improvements and a harbor in Neapolis, be 1-Turn-Worker-Factories on size 6.)
2. Be a Diplomat: Have everyone (except, maybe, Macedonia) in an alliance against Carthage and get big GPT-Deals by tech-brokering. Always have RoP with everyone. Give away 1 GPT to everyone. You don't want to fight anyone until you decide to do so. Trade maps every turn: If you have just one scout, you can make around 40 GPT by that.
3. Be a scientist: Research at 100% during the first 40 turns helps a lot to get big money from the AI. Philosophy helps greatly.
4. Take all luxuries you can so you don't have to waste coins on the luxury slider. (Use your first 4 Settlers for that: 1 for Spices in France, 1 for Silk in Hungary, 1 for Furs in Germany (if you place a city on the hill near Waldegesheim, you don't even have to build a road, as the goth's will do that for you), 1 for Gems, Incense or Dyes in the orient. (I usually ship 2 more settlers east as soon as possible to get everything.)
5. Time your Golden Age well: It should be late enough to have most cities at max population, but soon enough to get Philosophy and to be able to hit Carthage before it builds up a big defense. I'd say, around 300 and 270 BC is the time for that. Use the GA to build up a massive force of Horsemen (to upgrade them later: It's worth to think about disconnecting iron, if you already have Military Training: You have lots of cash and no shields to waste), Catapults and Galleys.
6. After the Golden Age, it's time to settle. There are a lot of Locations with fresh water and bonus food in Germany, Hungary and Spain: cashrush Settlers, as soon you have switched to Imperialism (what you really should do as soon as possible) to settle the best places.
In my last game (on Deity), by 40 BC I backstabbed Macedonia, because I was already near the population limit and just needed more land: The temple of Artemis helped a lot. As they had all troops fighting in minor asia, I also conquered all of their european holdings in a few turns.
At the same time, I used a cash-rushed force of Heavy Cav to conquer 3 egyptian Cities. That took me to 20/50 by 5 ad. If Egypt already is beaten by Persia (you can prevent that with a single Unit east of Pelusium), you could also attack the Celts, and/or Goths to push yourself over the limit.
But that's just a matter of taste. Most important is the build-up phase so that you can spit out huge amounts of units in the Golden Age.
The Persians surely are strong, but that's not your problem: For the domination count it doesn't matter if they own the cities in minor asia or the greeks do. If you fear them, just place a single unit (or, if you are at war with them: an army) east of Byzantium. That will hold them back. But you can outexpand them easily, if you want.
If somebodie's interested, we could make a shared game.
@iggymnrr: The games were on deity. Here is a
link, for everybody interested (starts with rbc3a).