The UP Chair is picked from the top 6 or so players in the game; I forget if it's top overall or just most influential, and what the exact numerical limit is, but it's something like that. I am not aware of any way to directly influence the votes of computer-controlled factions and GCIII doesn't allow you to trade influence if I recall correctly, so the size of your vote is the only thing that influences the likelihood that you'll win a vote that you can directly influence, and the size of your vote is controlled by your influence relative to the total galactic influence (or more specifically, the total influence of everyone in the UP) if I'm not mistaken. There may be some logic in the game which causes computer-controlled factions to be more likely to vote for factions that they like, though I wouldn't count on getting votes that way. Therefore, to make it more likely that you'll be the next UP Chair, you want to be one of the top players and you want to be one of the players with the most influence.

UP resolutions can be seen by looking in UPResolutionDefs.XML in /data/Game in the install directory; brief descriptions of the resolutions in a slightly more human-readable form can be found in UPResolutionDefsText.XML in /data/English/Text in the install directory. The tech-locked UP resolutions are those which list a prerequisite other than TechTree in the <Prerequ><Tech><Option>(something)</Option></Tech></Prerequ> section at the end of each resolution definition, where (something) is either TechTree or the internal name of a technology.