Definitely use the ESP instead of the Strat.
I'd try a "middle" pickup config (so if you only have neck and bridge it would be using both) or maybe the bridge with the tone pot turned down a bit but that guy has a rather "round" kind of tone which implies the bridge pickup is in there (and I think I can even see that his toggle is in the middle). Unfortunately he also looks like he's got some nice active EMGs so not sure if you stock pickups will be quite as smooth.
I'd start with a "Modern US" amp sound (like a Rectifier emu, maybe a 5150 but only if you are stuck) through a 4x12 bottom cab (matched to the Rectifier emu). Have it set to the drive/distortion channel but don't have the drive cranked (that is a tight distortion but it isn' searing).
You want a thick but smooth distortion (no crackling or squealing) which hopefully you'll be able to get close to just with that. I forget what the cab micing options are like in GR5 but I seem to recall them being rather limited. If at all possible try to get your mic options to be more toward the center of the cone (maybe not right in the middle but definitely not the edge of the cone which tends to be sharper... but play around with it). Also try (if possible) a 45 degree mic and rear mic sounds. Listen to all three mic setups on their own to see how each is reacting to the sound and blend them to taste. I recently learned that 45 degree mics tend to give a better modern distortion sound (and I personally like how rear mics give some fullness to a tone).
The other thing I'm hearing in there that will be trickier to nail down is there is something really eating into the bite of the tones attack. It's almost like really fast and precise gate producing a quick volume swell for each note being played. This MIGHT be some fancy compression work but I don't know. I'd look for some kind of transient shaping effects in GR5 (or maybe even see if Sonar's Transient Shaper plug will work). Basically you just want to really smooth out the attack of the note so that you can't hear the abruptness of the pick/hammer on. You are going to have to be careful with it though because I find that type of thing ends up screwing up the parts of the notes you want to keep or the timing gets all screwed up so the effect delays/softens the attack TOO much. I have stumbled across this type of effect while browsing presets within sims or using multi FX units but I do not know exactly what is used to do it.
As to the "legato"... well he's playing those. Just hammer ons/pull offs. I can't think of any traditional guitar effect that would produce something like that and still sound natural. You may be able to fake it with some kind of fancy delay set up (so that the first note is held in a way it overlaps the second note and maybe slightly delays the start of the second note). Of course as you sped up and slowed down it might ruin the effect. hmm... maybe a slapback delay or something. It won't be the same as actually playing it but maybe it will get you close enough.
The otehr thing, and I have no idea if this would work, is put it into Melodyne. Then go in and chop the attack part of the second note off and drag it down to the pitch of the first note.
Essentially what the guy is doing is picking the note immediately BEFORE he fingers it then hammering on/pulling off to the actual note. Not a particularly difficult technique but he is doing it so fast and consistently and timing it all very well that it's creating that interesting off time staggered effect. I like it.
So if you can duplicate that in Melodyne by attaching all the pick attacks to the end of the previous notes and make sure you have Melodyne set up so it doesn't hit the start of those second notes with a hard attack (I'm pretty sure there is a variable setting for that) then it might work.
The problem is performing it in such a way to anticipate the slurs that won't exist until you create them. It might be tricky to time it... however you may be able to do the Melodyne edits then nudge the clip on the timeline to stagger the notes like he does.
Another thought that comes to mind is you might be able to feed the audio into one of your fancy synths and access a legato setting. That way the synth is processing the audio and doing the heavy lifting for you. Again I have no idea if that would work and I don't know much about synths but I have seen some pretty interesting stuff done to audio this way in various vids and tuts.
Anyway... just some ideas I had reading your post and watching the vid. Maybe they'll give you some things to experiment with to get you close. I am NOT a pro engineer (but am a "pro" guitar player... even though I haven't been paid for a while but that's just because I ain't sellin' at the moment).
Cheers.