If you go to the government page it'll show you who is next in line to be consul/archon/whatever. If you mouse over the sundial it'll show the top 4 candidates for the next election based on the number of votes they'd currently get. Go to the character page and mouse over the Succession Support number (below the statesmanship bar) to see a breakdown of where those votes come from.
The tl;dr of it is that it's mostly prominence, popularity and party. Increase a character's prominence & popularity through jobs and war, increase their party representation in the senate by getting a good censor (make sure the censor is part of the civic faction) and party leader (this is random, but get them killed or bring them to trial if they have super low charisma), or through that thumbs up button for a little tyranny. you can also nudge characters towards certain factions by doing things like making them general (for the military faction), or granting them holdings in the case of the civic faction.