Diplomatic, UN or AP, is the easiest win for me to achieve. Under 3.13, you have to make friends so no cheesy early AP win (which is the fastest win other than a tiny-size duel conquest).
The key requirements are: build the AP/UN so you control it -- though that isn't an absolute requirement. If you don't build it, though, you should be #1 in votes.
Getting to be #1 in votes is just a matter of expansion. There are lots of strategies to pull off that, but conquest is generally a good one.
The last trick, and it isn't so hard once you know how, is to make friends. If your friend asks you for something, give it -- unless you'd be seriously harmed doing so. Even then, consider whether the negative diplomacy is an even worse problem.
Change civics and religions to establish alliances and trades. Once you have enough positive modifiers, you have a fairly good shot at getting the vote to win from an AI.
One more tool for winning is the mutual struggle modifier. If your opponent in the vote is at war with you and your allies, this boosts your relationships with your allies.
I think that the diplomatic victory is easiest because the conditions to achieve it come up often in normal play sooner than the others. The AP is way, way earlier than space race, and even the UN comes along sooner. Domination/Conquest requires that you can beat everyone. On many maps, this just isn't practical early in the game. You need enough size and a tech advantage in order to pull it off. Diplo doesn't require superior technology, though it can help. I've won UN victory with way inferior technology, by establishing an alliance against the tech leader for the votes.
Cultural can be easy with the right setup but still takes time, and the AP comes along sooner. The AP is actually pretty good for a cultural win attempt, as if the diplomacy fails you can still win without being huge or having a major tech lead.
Space race is pretty easy if you can hold a tech lead but what if you don't? The only way to win is to play catchup in tech, or beat down the leaders in war. Neither is easy.
Successful early warmonging helps out most strategies. There is nothing like having two civs worth of empire and a rival taken out to give you a commanding lead in the game. But domination/conquest gets harder when the travel distance is greater. You need the right techs in order to truly handle a global empire. Otherwise, you get what I call the Mongol crunch. I do it often as Ghengis -- conquer a lot, then struggle on building wealth and perhaps pillaging in order to stay solvent.