Why does Claudius not proceed with legal action against Hamlet for killing Polonius?
Most likely because Hamlet is the prince, and therefore above the law in some senses. There are a few other possible reasons too however. Legal action would have been too 'by the book' for Claudius. In other words, Claudius is a murderer - he murdered his brother. He deals with things in a different way than most kings do. Also, he is beginning to know that Hamlet is a threat to him so he would rather do something that ends in Hamlet's death. Legal action could have taken too long. And finally, Claudius is unsure like the rest of them about Hamlet's sanity, although he suspects Hamlet knows what he is doing. So, he sends him to England with Rosencrantz and Guildenstern and a note that tells the King of England to behead him. Hamlet of course, escapes.
I believe another reason Claudius gives is his "love" for Gertrude and how it would crush her to have her son tried (and possibly executed) for murder. He decided it would be better for her if he just sent Hamlet away to England, making sure that Rosencrantz and Guildenstern deliver the letter to the English king, ordering Hamlet's immediate execution. At least then, Gertrude wouldn't perceive Hamlet's death as being tied to Claudius in any way.
In addition to both these fine answers, Claudius knows something must be done with Hamlet and that something comes in the form of Polonius' son. The two of them concoct a duel, in which Hamlet is poisoned by the tip of his opponent's sword. All this is agreed upon after Hamlet shows up again from England having escaped from that diabolical plan. Hamlet does succumb to the poison sword, but not until after most everyone else is also dead.