While reading some books that discuss the criminal prosecution process will help you learn about it, that learning would be more for your personal edification than anything else. Only an attorney may represent him in any legal attack of his conviction. Moreover, it'll take a good appellate attorney to spot the potential errors in the case that might at least get him a new trial. Appeals and collateral attacks of a murder conviction are complex matters for experienced appellate attorneys. The prosecutors will have their best lawyers working to keep him in prison. If he has any real chance of getting the conviction set aside, it'll take a good defense lawyer to go up against the prosecutor.
In a capital murder case the defendant an indigent client (which is nearly all murder defendants as few people could afford to pay for what a murder defense would cost) is initially respresented by an attorney of the public defender's office. But in an appeal of the conviction he'll likely have a different set of lawyers for that, especially If there was any indication that his trial team as a whole provided incompetent representation. If he got a different lawyer for the appeals that lawyer would have looked at the possibility of a challenge to the conviction based on incompetant representation at trial. If there was something to pursue on that score, the appellate counsel would pursue it. It may have already been done and rejected by the courts. If that's the case, then what you are trying to do will be a dead end. If he has lawyers still working on appeals or other legal maneuvers he should not be discussing his case with you at all. Doing that destroys the attorney-client privilege and makes you a possible witness against him in future hearings or in a future trial.
If he had at least one competent lawyer on his team and that lawyer directed his defense then another attorney on his team falling asleep is very unlikely to make a difference. He'd have to prove the defense team as a whole provided incompetent representation and my guess is that it will be very difficult for him to prove that. Convicts sitting in prison on a capital murder conviction with a possible death sentence looming over them are going to grab onto anything at all that sounds like it might help them get out of prison and hold on to that hope very hard even if, objectively, what they have isn't all that much use to them in challenging the conviction. They need that hope to help manage the depression they tend to suffer in prison with the prospect of never leaving that place alive.
Also, prisoners have been known to spin tales of woe for people on the outside to get things from them. In other words, they scam people by playing on their sympathy.
So be careful. You don't want to be a scam victim. And if he's telling you the truth, you don't want to be the reason the attack on his conviction fails. If his lawyers don't know you've been communicating with him, they need to know that to cut off any future problems that might arise and to do damage control for anything already done that might hurt him.
I don't mean to sound harsh. Your heart is in the right place by wanting to help others. But its important for you to know there is little that you can do for him in attacking his conviction or sentence. His lawyers have to do that. There is, however, the chance you might unintentionally make things even harder for him. Before communicating further with him, my suggestion is that you talk to his lawyers to make sure you understand the things you should not discuss with him. Bear in mind that all prison communications by inmates are screened by prison officials and recorded, with the exception of protected conversations with his lawyers.