Playing the game more like a Domination Victory will likely lead to you getting enough votes to easily win with the help of just one voting buddy, who could even be your vassal.
Assuming that you don't play the game too militaristically and thus will have to rely on votes from AIs for a good percentage of the votes, then you'll be up against the AI with the highest population in the voting. So, your voting buddies should be successful in growing their Civs, but not so successful that one of them becomes your voting opponent.
The required AIs are not the easiest bunch to work with. Hatty tends to be a good voting buddy since she has Organized Religion as a favourite Civic and she gives a decent bonus for a shared Religion (and having a Religious Civic keeps her from easily abandoning her State Religion for Free Religion, meaning that you can pretty much rely on getting the shared State Religion bonus from her if you share her State Religion), but that Favourite Civic conflicts with Zara's Favourite Civic of Theocracy. On the plus side, Zara tends to expand really well, so he has pretty decent odds of becoming your voting opponent.
If necessary, you can vote everyone into running the Environmentalism Civic, in order to please Sitting Bull, although then you won't be able to run Mansa's Favourite Civic of Free Market. Mansa is likely to tech well and thus get to Economics for Free Market before you can build The UN, and Mansa will be likely to still be trading techs with you at that time, so it shouldn't be too hard to get into Mansa's Favourite Civic.
As lymond suggested, try picking 1 to 3 additional AIs whom you feel are relatively easy to get as voting buddies but who generally won't expand so well as to become your voting opponent. AIs who will easily trade techs are probably good choices, so that you don't end up with AIs who won't trade you a tech even if all of the AIs on the same continent know said tech.
If you're not very good at warring, do your best to domestically spread the Apostolic Palace's Religion and see if you can use it to help you out with warring, either enforcing peace or getting some AIs to gang up on another AI, in order to get a Mutual Military Struggle Diplo bonus. That said, watch out for the AIs liking each other too much, so you may have to bribe some AIs out of a war. I believe that the Mutual Military Struggle Diplo bonus goes up every 8 turns, so you could have a war ally for several turns and then bribe that war ally out of the war by the 7th turn so that the war ally that you bribe out of the war won't get a Diplo bonus with the other AIs that are fighting the war on the same side as you.
With the game being a Continents game, it may be worth looking for a map where you start with the right types of AIs on your continent. For example, if you're good at warring, you might want the AIs whom you want as your voting buddies to start on another continent, so that you can plan to take control of your continent and then play nicely with the AIs on other continents.
I'm not sure if it's possible to be able to meet AIs on other continents by using just Galleys with the right map, but if you can get such a map, then you'll have more time to make friends (and make enemies) if you can meet all of the AIs prior to learning Astronomy. Otherwise, beelining Astronomy could be a good play, so that you can start mapping out the diplomacy web relatively earlier in the game.
From my understanding, Diplo games are best done with a lot of Great Scientists, so try to allocate a couple of Cities early on to focus on just hiring Scientist Specialists for quite a long time. The investment takes a while to pay off, but when it does, you'll find that Lightbulbing a tech is what may allow you to trade your way out of a tech hole.
Some people argue in favour of generating a Great Engineer in order to complete most of The United Nations. My take is that you just want to found a City that would make for a good National Park City, but instead of planning on building a National Park there, you will Chop the Forests into The United Nations and will whip other build items to overflow their excess Hammers into The United Nations, meaning that you can just stay focused on generating Great Scientists. Don't plan to rush-buy The UN or to directly whip The UN, since The UN has an extra cost on top of most other Wonders for doing either of those actions and you're better off just whip-overflowing other build items into The UN, whether you sit down to do some math and chain-whip a cheaper build item into a bit more expensive build item into an even more expensive build item where your whipping allows each build item to overflow 25+ base Hammers or whether you're just whipping a build item and overflowing into The UN on the following turn repeatedly for several turns. A good National Park City has Food Resources as well as Forests, so you should have some population points to whip in that City, too, right?
