I have won the scenario with Carthage (deity, no cheats).
I think, there were 5 pillars to success:
1. Carthage has one big problem in the game. That are the roman legions. The question is, how to deal with them. And the answer is: Dont.
You have to destroy the roman supply of iron. Rome has only two of them, one near the city of Rome, one near Neapolis. So one should use the initial army (filled with swordsmen and/or Numidean Mercenaries) to cut them off. One can stay with the army in Italy, if one wants to. I would suggest, not to destroy all of the roads, since one will need them for quick movement later. (Its somehow dirty, because the AI never seems to attack an army with a green hitpoint-bar. I can live with that.)
Thats the most important key for carthagian victory, I think. Since Garrisons also need iron, Rome will fight mostly with Spears and Archers for the rest of the game.
2. Maintain naval superiority. Sink the initial roman fleets and build galleys on Sicily and the other Islands. Place one galley near every roman port so that you can sink every roman fleet.
Later, it's useful to blockade the western mediterranian with 6 ships to keep persian and macedon fleets outside.
3. Another crucial factor is diplomacy. In the first turn, you can get everyone into an alliance against Rome (plus nearly everything they have. Since the AI would not accept such deals in other turns than turn one, I think, its an exploit and would suggest to set yourself a limit). Use that! Before you have built some Heavy Cavalry, it's not worth to attack anyone anyway. It's better to let the initial roman Legions disappear in Gaul, Germany and Greece than seeing them in Spain.

I made ROP-agreements with everyone. Only after having Rome on their knees, I began to prepare for an attack on Alexander (since he nearly always succeeds to build the Temple of Artemis in Athens: And that's important, because you want your borders to expand quickly).
So: Be nice with everyone, trade around maps to make some gold. As soon as you meeet the Barbarians: Give them all the techs you can give them: They'll get them anyway. But you want to be the one to get the return!
4. Technology. Carthago can easily reach Philosophy first, when it wants to. And it wants! The free tech will be incredibly high-cost Monotheism. Nobody else will research this useless tech, but everyone will pay a lot for it.

After that, go all the way to imperialism. AI will normally research the rest. You can trade away the techs for money, if the others don't have techs to trade. I managed to have around 10.000 Bucks in treasury when I switched to imperialism (what you should do! With that short tech-tree, you need investment opportunities for your money!). It's imortant to always research on 100% though.
5. Luxuries. If you are filthy enough to get yourself Khmun in the first turn in exchange for helping the poor egyptians against rome, you'll have no problem to rush a settler there to place a city near Silk and a Colony near Incense. If you're not filthy enough, you should nevertheless try to get those egyptian Luxuries. I would suggest to send settlers as early as possible. Also try to get the Gems on Sinai sooner or later. In addition to Spices, Wine and Dyes from Persia, (later out of Lybia) that's enough to keep your small towns happy. Later, send a galley with a settler and a worker to Scotland to get the furs up there. You should try to get all your distant towns to party, since that gains you a lot of shields.
One big advantage, that Carthage has, is, that is doesn't have to waste his Golden Age. Instead, you can wait for your cities to grow and for your workers to develop the lands, until you decide to win your first battle with war elephants. The golden Age should then be used to buil up a really big army of Heavy Cavalry and Catapults. With at least 10 of them each plus some Numidean Mercenaries, an Invasion on Italy has a good chance of success. Starting in the south seems most reasonable.
Without Legions, Rome is quite easy to kill. Hopefully, some Leaders popped up too: Now you can pick your next enemy. Macedonia will be the best target, I think. Take care, that you get them with all their wonders, before the persians do so!
By then, with having done some colonization in Spain and northern Africe in the meanwhile, you should be on the road to victory.
I had about 30 turns left, when the conquest of some egyptian cities finally got me those 50% of the world population.
That's my strategy for Carthage.