If anyone steals from Solaf's shop, he may send hired thugs after them, with a contract stating that they need to be taught a lesson and possibly killed. This may happen even if he didn't see the theft.
Apparently Solaf is courting Tekla, although no mention of such is made through in-game dialogue. When the Command Console or Creation Kit is used to examine Tekla's relation towards other characters, it is revealed that Solaf and Tekla are courting.
He sends the Dragonborn a letter if his rival is killed.