Every semester there are elections held for the open seats on the Senate. In order to run in the elections, the first person to talk to would be the Vice President of Academic Affairs and Governance - if they are not managing the elections that semester, then they will find out who is first.
Either way, no matter who it is, there are some pretty specific guidlines as to how the elections need to happen. Currently, they are listed in Article XIV of the bylaws as well as Article XII of the Constitution and Article X of the Constitution. I suspect the rules that will exist, at least until there is a rewrite of the Constitution, would still be located in these places.
I also know that there was discussion of the political parties last semester, so I am cautious about stating the rules here since they are likely to change soon.