A good starting point is a book written by Micheal Naughton - Claims of Innocence - An introduction to wrongful convictions and how they might be challenged.
A guide to the appeal system can be found here
Why might a defendant appeal against his / her conviction?
If someone is appealing against their conviction, they will be asking the court to quash the conviction because it is 'unsafe'. There are various reasons why they might claim the conviction is unsafe and the following are a few examples:
New evidence has come to light since the trial which, if it had been available at the time of trial, would probably have meant the appellant would not have been convicted eg scientific advances in use of DNA evidence
The judge was unfair in the way she/he summarised ('summed up') the case for the jury or the judge didn’t follow the written guidance which is provided for all judges so that they summarise legal points correctly for the jury. The written guidance is called the Crown Court Bench Book - Directing the Jury
The judge made a decision of some sort during the case which was not fair or was not correct in law eg about whether a particular piece of evidence could or could not be mentioned to the jury.
The normal procedures for ensuring a fair trial weren’t followed, which might mean the conviction isn’t safe.
Centre for Criminal Appeals are a small charity who support victims of unsafe convictions