Someone can probably explain this better, but I'll give it a shot. Presuming you already know how to generate tourism (and i presume culture too
). Each turn, if you produce 8 tourism per turn for example, that 8 tourism is added to several meters, one for each civ you know. This meter competes with a similar meter they generate, which is their culture. So, if your total tourism (influence) you have over them surpasses their total culture (not the progress to a social policy the amount of culture they have got throughout the entire game), then you become influential on them. If you become influential on everybody you win. You do not need to be influential over wiped out civs.
Right, World Congress. The first person to discover the printing press tech AND discover every civ in the world founds the world congress. Each 'session' lasts about 30 turns. At the beginning of each session two civs propose a resolution. At the end of the session (i.e. 30 or so turns later) everybody can vote in the congress. Each civ has a certain amount of delegates (if you are the host, i.e. if you found the world congress you get extra votes, and the more city state allies you have, the more votes you get (this starts in the industrial era)). When voting begins you can assign your delegates to the two resolutions that have been proposed. For example, if you have 7 delegates, you can assign all 7 of them to vote 'yes' on a resolution, all 7 to vote no, or mix them in between (i.e. 3 yes on one and 4 no on another.). Every civ does this. If a resolution passes, then its respective effects are in (no pun intended) effect(i.e. ban luxury sugar means nobody gets a happiness bonus from sugar). Another proposal you can make is to repeal resolutions, setting up a vote to remove their effects.
Lastly, I should mention the special sessions. Every time half of the world reaches the next era (starting with the industrial), a 'host vote' is called. You have the choice to vote for any player (including yourself), and the winner becomes the host, gaining bonus votes and the guaranteed ability to be able to propose every session. The host and the player with the second-most votes proposes. Also, the world leader proposition. Same as the host vote, except it starts when half or more civs reach the atomic era and reoccurs every set amount of turns (10?). The winner of this vote wins the game (although there is a quite high minimum number of votes required for your win (in the vote) to count for a win in the game. The two top two civs that receive the most amount of votes get bonus delegates for later on.
I've probably missed something but I hope this helps
.
EDIT: Yes sprawling culture can work. Have a look at the very attractive 'holy sites' religious belief.