This is a very interesting question. He apparently had two terms in the senate, got quite reasonably sick of that institution's legendary intertia, and quit. (He served out his two terms but did not run for re-election, which he would have almost certainly won.)
I would feel a lot better if he had been mayor of a major city or governor of a major state. Governor's probably the best position since most states are organized like the Federal government in miniature, and so they give the candidate a good idea of what he will be up against.
I'm dismayed at how few candidates in this race have successful executive experience, to tell the truth. Aside from Rudy, the closest match we have is Hillary Clinton's period as "co-President", where she pretty much flopped.
I think the reason there is so much interest in Thompson is that he is more socially conservative than the leaders, and has the name ID to make a difference in this race. For that reason, it's much harder to dismiss him as someone who cannot win, like Duncan Hunter. (To give you an idea of this, note that Duncan Hunter has never had over 2% of polls, while Thompson is at 9% with no campaign organization yet.)
From what I can see, Thompson's acting career was considerably more successful than his political one. Nothing wrong with that, but it doesn't make me think of him as Presidential material.
D
He quit for a VERY GOOD REASON- his daughter was dying and he needed to be with family. Get your facts straight.