How do you play a trill fast without the sound fading away? I understand that flicking the note you are pulling off from is involved, but it seems impossible to flick and do them fast at the same time. I'm trying to play the Trooper, and I can play the trills consistently fast, but the sound always seems to fade away.

Pick the first note, hammer and flick.
If it's not working up to speed just practice the flicking motion slowly.
also helps if you have some gain although it's not necessary.
On playing the Paul Gilbert signature at the guitar store extensively, my missus sighed:
"Put it down now, It's like you love that guitar more than me!"
In Which I replied.
"Well it has got two F-Holes!"
Trills are basically Hammer-ons and pull-offs if I'm right (LOL what have I've been smoking)
but if you dadshadsh
just sloow down and do the hammer on and pull off, be sure of what your doing.
Then speed up to the timing on the song. Trooper's Iron maiden, am I right?
Your trills might also be sucky because of a sustain problem with the guitar, like I have lol
It may be the guitar, because of its sustain, when I learned it I just had to hammer on and pull off really hard. I was taught to hammer on and pull off like this so I always have. Perhaps try and play it unplugged and see if you can still hear it after hammering on as hard as you can because then you'll know if you will be able to do it when amped?
My name is Jack.
Yeh I often get the same problem as you, I think its the sustain on my guitar, working on getting a new pick - up to increase it

I've also tried unplugging and trying to hammer - on as hard as I can.

I've also done it on my acoustic guitar as the sound is harder to make with the steel strings. I now pull the string away when I trill as a get a good pull off sound, Sorry if that doesnt make sense :P