I'm using a Squier Standard Telecaster, but the stock pickups are too twangy and weak sounding. I'm trying to get a more powerful and full sound. Think Jonny Greenwood of Radiohead.

Here's a link. The sound I'm trying to get is like the solo that starts in 3:46. Would I be able to get similar sounds with Duncan Hot Rails? Suggestions of other pickups would be nice too.

Note: Please don't say "Just buy a new guitar"
I don't understand why you can't just buy a Duncan hotrail - Like an STHR-1B if that's the sound you want.

I've never heard a Squier Tele pickup, so I don't know how bad they are (although if they are like the Squier strat pickups, quality wise, I might have an idea), my project Tele has '62 re-issue Jap pickups in, and they have a nice tone - still a very telecaster tone though. They can be found on Ebay fairly regularly.

It'd also be worth looking Kent Armstrong pickups, a TEHR-1 comes in at 13K so it's not quite as hot as the SD.
First off, what's your amp, what's your EQ?
It better be a tube amp, and it better be cranked.

All squier telecasters are pre-routed under the bridge for a full-sized humbucker, all you have to do is get a humbucker-sized bridge. there's one at: http://store.guitarfetish.com/chhubrforteg.html

As for WHAT pickup, I'd say go with vintage, because everything's pretty much based off of the vintage PAF humbuckers, the P-90s, or the Vintage strat pickups.
I put a combo of the GFS mean 90(click the link above then the GFS pickups tab, it's in humbucker sized pickups) in the bridge and a reverse-wound, overwound strat pickup in the neck. I have the biting aggressiveness and early rock/blues from the P90, and the tones of the vintage strat in the neck, as well as a rather unique humbucking sound when I run the two together (it would probably do that solo in the song you linked to rather well, actually). Very versatile, very simplistic, it fits right into what a telecaster should be.
Please do not insinuate anything sexual from that.

Quote by cobain_is_king
If your friends don't like your guitar, cover it in stickers and it'll be teh rawks!
Why not just buy the SD Hot rails? But if you do, make sure you have 500k pots. Else the hot rail can get quite muddy.
Empty tankard! Empty Tankard!
Because the hot rails is meh.

I listened to the video. Definitely think this:
Is better than the Hot Rails for what you want.

That's if you want to use your bridge for that. It'll give a fuller sound, less twang.

It sounds more like he's using his neck pickup, though, to me.
jonnys tele is a tele plus, meaning it has fender lace sensors instead of standard tele pups. both his teles have a blue lace in the neck and two reds setup as a humbucker in the bridge. if you can afford it, go with that.