Strictly in terms of the movie, Thenardier extorts as much as he can from Valjean initially, for the reasons that @Christian_Rau mentions (sees his wealth, knows he's willing to pay a lot). Later in life, their meeting comes not from any pursual of Valjean, but rather from chance. The Thenardiers are living in Paris, learn Valjean and Cosette have moved there, and plan to rob them (again, knowing how wealthy he is, and also out of revenge).
The scene with the ring is actually not really an attempt to discredit Valjean, but an attempt to blackmail Marius -- Thenardier is hoping that Marius will pay him well to keep the secret about Valjean "looting" bodies in the sewer. It backfires, though, as it's the piece of information that lets Marius know his father-in-law saved his life.