If you want to be able to more easily overcome the unhappiness penalties associated with having a greater number of cities, you might want to try playing as Egypt. Their unique building, the Burial Tomb, replaces a building that you were likely going to build anyway (the Temple), and only requires a Monument already in place to build it. If you time Legalism properly, you can even get them built for free in your first four cities. It has no maintenance cost and provides an extra +2 happiness, making the per-city unhappiness penalty much easier to manage. You'll also probably want to keep cities relatively small if you're going to build a lot of them. I recommend looking up some strategy tips for running an ICS (infinite city sprawl), since I've never tried it myself and can't really give advice on that front. Similarly, Persia's UB, which replaces the Bank, also provides +2 happiness, but building a Bank in each city requires a bit more effort.
Egypt's UA can also be helpful in combating unhappiness, helping you to hard-build useful wonders like the Circus Maximus, Taj Mahal, Notre Dame, and Forbidden Palace more quickly. Policy-wise, you'll want to target Meritocracy (+1 happiness for each city in your trade network, which should be every city), and Organized Religion, which grants an additional +1 happiness for the Monuments and Burial Tombs you'll already be building everywhere. Altogether, this can net you +5 extra happiness per city, negating the per-city modifier and the first two population points completely.
