Ok, I'm trying to learn Closed Casket Requiem by The Black Dahlia Murder, and it's going pretty well, except the string skipping is giving me a problem. Is there a trick to it, or do I just have to build up to doing it fast? The chorus part in Confined by As I Lay Dying also gives me problems like this.
I used to have trouble with string skipping riffs. Its just something you'll have to build up slowly over time raising the metronome a little bit every time you get perfection. Titling your right hand slightly downwards to anticipate the string skip might help, tho make sure you are still correctly palm muting while doing that. Try playing around with the positioning of your right hand and how the pick hovers above the strings. Make sure your wrist is loose and your fingers are not holding on or anchoring on to the high strings because this can cause a build up of tension that you don't want. This can help you a little bit but its not gonna give you an amazing burst of speed ability.

But just as a side note, if the chorus in Confined is giving you trouble (i know exactly the parts your talking about), then Closed Casket is gonna give you a huge amount of trouble because its a signifcantly faster piece. The only thing Confined has above the opening to Closed Casket is there are multiple root notes muted whereas Close Casket just mutes the open C.
I'd say get Confined down to perfection before you tackle any Black Dahlia stuff.
When i'm down picking higher note, should I alternate pick the open C, or down pick it as well, which would probably be easier once I get used to it?
Learn it slowly at first, play it at your speed until you get fast. Then play later on with the track. This song was a biznatch when I was learning it for a school cover.
I usually alternate pick the open C for that kind of speed (down pick note up pick root note), its easier for me. Not sure how BDM does it tho.