For this kind-of logic problem I would first figure out if anyone is telling the truth, or if everyone is saying that everyone else lies. When I worked out this problem I set up a grid on a sheet of paper listing A-E and marked when someone said that they lied.
From there, if there is someone that is only truthful, then you know that what they say is right about another person. You can then judge the truth of what other people say based on the truth-teller's sentence. For example:
If I know that Hector tells the truth, and Hector says that Robbie lied when he said that Tommy was lying, then I could conclude that Tommy actually tells the truth because Robbie lies. All based on the fact that I know that Hector tells the truth about Robbie.
From there you can determine which people lie and which tell the truth.