The Silence traveled back in time to Series 6 to bring up River Song and kill the Doctor because they didn't want him coming to Trenzalore? But why is that? I know about "Silence must fall when the question is asked" but they help the Doctor fight the Daleks in 'The Time of the Doctor'. So is it that they wanted to kill him before he came to Trenzalore because they didn't want the Time Lords back into the universe, or is it that they wanted to kill him because they wanted to protect the inhabitants of the town Christmas?
This is an ontological paradox, or a time loop, really. Like the Doctor said in Series 9 Before the Flood,
Basically, in an ontological paradox the cause becomes the effect.
In this example, The Silence was actually a "splinter cell" that broke away from the Papal Mainframe, according to The Time of The Doctor. They traveled back in time to prevent The Doctor from ever reaching Trenzalore, but in the end, they became the very reason the Doctor ended up there. From the script of The Time Of The Doctor:
TASHA: Why did you ever come to Trenzalore?
DOCTOR: Well, I did come to Trenzalore, and nothing can change that now. Didn't stop you from trying though, did it?
TASHA: Not me. The Kovarian Chapter broke away. They travelled back along your timeline and tried to prevent you ever reaching Trenzalore.
DOCTOR: So that's who blew up my Tardis. I thought I'd left the bath running.
TASHA: They blew up your time capsule, created the very cracks in the universe through which the Time Lords are now calling.
DOCTOR: The destiny trap. You can't change history if you're part of it.
TASHA: They engineered a psychopath to kill you.
DOCTOR: Totally married her. I'd never have made it here alive without River Song.
In the end, The Silence ended up causing precisely what they tried to prevent: The cracks in the Universe the Timelords were trying to come back through. Kind of ironic, if you ask me.
To clarify: The Timelords were trying to come back to the universe using the cracks in time caused by the Universe blowing up all the way back in Series 5. However, they knew that there was a risk of the Time War starting again if they did, so they deployed the truth field, and started asking for The Doctor's name (which only The Doctor himself knows, at least according to them, since he's the only Timelord left in this Universe), which would be a signal from him to tell them it was safe to come back. It wasn't, because the Daleks were already waiting for them ready to open fire. The Silence tried to prevent this stand off by preventing The Doctor from reaching Trenzalore, which not only failed, but ended up causing the whole thing. In the end, when the war started anyway, whatever was left of The Silence joined the Doctor to stop the Daleks from destroying the town of Christmas.