The Thalmor don't want the Civil War to end, as it weakens both the Empire and Stormcloaks, furthering the Thalmor's goals without actually doing a great deal. Also, it's likely that they'd prefer the current Emperor to remain, as he was the one who agreed to the White Gold Concordat. The Thalmor need longer to recover from the Great War, as the mostly-human Empire can regenerate itself much quicker, and a new Emperor may have a more aggressive view on how to approach the Thalmor threat.
It's pretty hard to RP a Thalmor, as there aren't a great deal of quests for you. However, if you align yourself with the Thalmor goals insofar as tracking down Talos worship, that's simple enough to do. You could also wipe out travelling soldiers or secure well placed camps, etc.