Jeremy's case is rather unusual compared to most. If Jeremy is guilty his motives for protesting his innocence there are several possibilities.
1. He knows the prosecution has been corrupted and evidence tampered/manufactured because he knows exactly what he has done and how he done it. His biggest dilemma is the try and prove it without incriminating himself. Him saying for example "The silencer is false evidence because I never used it when I killed them or I buried the silencer I used" will prove the corruption but defeat the object. So the flaws that he is aware of could motivate him to try and crack his conviction.
2. Major aspects of the incriminating evidence against him can be cast into doubt
a) Julie Mugfords dealings with the police indicating possible coercion in exchange for immunity.

b) The extended family who came forward with the evidence against him stood to Inherit the Bamber estate if he was convicted. In todays money worth £1.3 Million Pounds ($2,000,000) giving them possible motive to corrupt or falsify evidence.
c) The police investigation was flawed (however to be fair Essex Police had never had a situation like this before)
The above 3 points don't prove innocence by any means but does harm the credibility of the evidence against him. This will motivate him to continue to fight for freedom
3. One other possibility is that Jeremy is oblivious to the fact that he done it. His subconscious mind may have emotionally repressed most the memory of that night as a form of psychological defence mechanism to protect him from reality. And he now only has false instilled memories that come from the defence version of events. In other words he is living on planet Jeremy literally thinking he is innocent!