I would say that Stephano and Trinculo gain power over Caliban without even trying.  Once they see that they have power over him, they try to keep that power by playing on the fact that Caliban is really pretty ignorant and the fact that he hates Prospero.

When Caliban sees the two men, he thinks that they are gods.  This is especially true after they give him some alcohol.  He does not know about alcohol and gets drunk.

Once he is drunk, Stephano lets him think that Stephano is a god and that he and Trinculo will make it so Caliban will no longer have to be Prospero's slave.

This is all found in Act II, Scene 2.

In The Tempest, how do Stephano and Trinculo try to gain power over Caliban and are they successful?

Stephano and Trinculo meet Caliban after the storm causes the ship to come ashore.  Trinculo hides under Caliban's garments while Stephano approaches.  Caliban initially thinks that the two are trying to torment him; however, he sees that they are not.  Caliban tells the pair that he is governed by a "tyrant" (Prospero), so Stephano and Trinculo use this against Caliban to attempt to gain power over him.  They tell Caliban that they will break into Prospero's castle to kill him while he sleeps and that then they will take over the island with Miranda as a "prize."  Caliban is enticed by this; and although Trinculo thinks it is funny that Caliban would worship a stinking drunk, Caliban is so oppressed by Prospero that any other master seems better to him.  Caliban does end up following Stephano and Trinculo and pledging his loyalty to them--he promises to show them the secrets of the island in exchange for their taking down Prospero.