Serious question/ what parts make us think Avery didn't get a fair trial? If the evidence is tampered with or whatever then that sucks, but my understanding is that many appeals center on improper counsel (he had great counsel) or an unfair trail. It doesn't seem that either of those apply here.
Let's assume that the blood and other DNA evidence was planted. There wouldn't have even been a trial in the first place. So the fact he was on trial was an unfair trial. At best, it would be reasonable to assume a crime occurred there, but there is no evidence linking anyone specific to the crime.
They have no proof or even a bit of supporting evidence that the evidence was planted though. I would think the courts would need something that supports that concept before they could consider an appeal. Common sense screams that the cops planted evidence. But if the jury bought it, and there is no new information, I'm not sure the legal system has a path to a new trial for him.
The proof/evidence is that they were discovered by people who were determined to have a conflict of interest, who have suppressed/tampered with evidence in the past that would exonerate the defendant and should have been nowhere near the crime scene to begin with. That's 3 strikes.
That was for the jury to decide though. There was no legal sanction against them being on the crime scene and finding evidence. They voluntarily removed themselves from the investigation, but then didn't. I am the farthest thing from a lawyer, but what I am trying to say, is that even though to us it looks like there was impropriety, from a legal perspective there wasn't any. The defense had the opportunity to impeach the testimony and the evidence presented to the jury, which they did, and they failed at convincing them. From an appeals perspective, there has to be either new information or evidence of impropriety within the trial itself.
A lawyer should probably answer this for real, but I can understand why the appeals were denied. Avery's defense team was actually quite competent and nothing new has been uncovered since he was convicted. The jury seems like they had a horrific interpretation of what was presented, but that doesn't get you an appeal.