Or if you want to have a lot of cities (not puppets) play France.
You cannot win a culture victory with more than about three core or annexed cities. France is not your best civ for a culture victory.
Siam is your civ for a culture victory. Ally with cultural and maritime city states to get the double culture and double food bonuses respectively. You'll be tearing through policies. The food bonus will help you to grow to truly large populations with no happiness issues. The key is to micromanage your population.
Great artists are your friends. You can expand your borders to capture key resources or just to give you some separation between your city and potential enemies.
Great engineers can help you capture the key wonders. Here is a post from another subject:
I would start an OCC with either SIAM or India. India is intuitive for it since Gandi's passive benefit punishes expansion and rewards large populations. SIAM isn't so intuitive, but it's UB and passive benefit allow you to blow through policies. The Wat, which replaces the University, also provides +3 culture. The passive benefits doubles city-states culture and food. You will rack up culture and grow a huge population. You will also never experience unhappy citizens.
Policies: Tradition, Piety, Patronage, Freedom, Order
Build your one city on a hill for the defensive benefits, and keep some up to date units for self defense. Keep an up to date siege unit in your city. and some modern melee units in defensive positions. Prioritize "The Great Wall, "Hemeji Castle," and "The Kremlin" for defensive purposes. With walls and a castle, you will be practically impregnable.
Get a few units out early to get the cash from barbarian emcampments and scouts to search for city-states. If you are on the water, get a Tireme out early for the same reason.
Great artists are your friends. Not for their cultural value, but because they can expand your territory tremendously via culture bombs. Have them expand in the direction of luxury resources for happiness and trades. Even if you cannot "work" the tile, once improved, you'll still get the five happiness and ability to trade the resource. The extra territory will give you a lot of time to maneuver your forces to respond to an attack as enemy units move slowly (The Great Wall) towards your city. Once they get near, you hammer them with siege and have your couple of melee units mop up. Prioritize culture and policy-related wonders. Micromanage your specialists to pop Engineers (for wonders) and Artists (for land). Switch between different city focuses based on your needs. When you need to pop out a major build, switch to production focus, which will also generate great engineer points.
City growth is your friend. You will be able to run your economy on specialists AND working your most productive tiles. The "Statue of Liberty" is huge since you will be using many specialists as your population grows.
Prioritize "Stonehenge," "The Oracle," "Angkor Wat," "Cristo Redentor," "Sistine Chapel," "The Sydney Opera House," "The Hagia Sophia," and "The Louvre." Obviously, you cannot get them all, but you can probably get most based on your production capability.
This should get you started. If you want to make it easier on yourself, try an archipelago style map. You are much less likely to get attacked early even on higher levels, and you'll have time to implement your strategy. The other civs won't avoid you forever, so keep a good defensive force no matter where you start.