This character is about what you'd expect from a guy named "Wormtongue": he is a total liar. Like most of the villains in The Lord of the Rings, Gríma Wormtongue doesn't have a lot of depth to his character. He is just bad, without any back-story. But his badness takes a slightly different form from, say, Sauron's.
First and foremost, Gríma Wormtongue is weak. He likes to hide behind bullies who are bigger and stronger than he is so that he can increase his own power. Basically, Wormtongue is the Lord of the Rings version of Harry Potter's Wormtail (and we can't help but wonder if Rowling got that name from Tolkien). They are both low-down, cowardly, slimy guys who attach themselves to Bigger Bads for power. And it doesn't really work out for either of them in the long run.
The particular bully who Wormtongue hides behind is Saruman. Wormtongue does Saruman's bidding because he thinks it'll get him what he wants: Éowyn's hand in marriage (blerg!). So Wormtongue works to wear down Théoden in every way that he can. Weirdly, most of Wormtongue's evildoing happens before we even meet him in The Two Towers. We only see Wormtongue in person once Gandalf shows up to boot him out of Rohan. We know what Wormtongue has done, but in the novel itself, we only witness Wormtongue's punishment.
Wormtongue arrives at Isengard just in time to see his boss Saruman get his butt handed to him by the Ents. Wormtongue has nowhere else to go, so he joins Saruman in jail in Orthanc Tower. And just when you think that Wormtongue's life can't get any more pathetic, it becomes clear that Saruman is hitting and emotionally abusing Wormtongue inside their shared prison. Yikes.
When Wormtongue throws the palantír down at Gandalf, trying to hit him, Gandalf comments that he's not sure who Wormtongue wanted to kill most: Saruman or Gandalf. He hates them both equally. Wormtongue may be a pretty gross character, but by the end of The Two Towers, he's trapped, powerless, inside a tower with a man he hates, too scared to try to break free. Now that's a pathetic outcome for a man's life, even if he is a villain.