What type of evidence is used to show that the violence at Columbine High School might be related to mature video game violence

There is no real evidence to these accusations. They assume that some teenagers and children are violent because they play video games. The thing is, what teen/child DOESN'T play video games? You get a few sociopathic teenagers and politicians decide to blame it on video games.

Added:

The above answer is basically correct. Correlation does not equal causation; except in social psychology and sociology. The correlative arrow can point both ways here. It may be that instead of violent video games instigating violent behavior some people that will exibit violent behavior anyway are also watching video games. Scientifically we just do not know.