h = Hammer On


So when you are picking this (The first note on each string). Do you use alternate picking or a sweep like downward motion?
Which would be the better way for getting speed?

I would alt pick for comfort, but to build up speed as a practice run, downpick each one in a sweep.
call me ziggy.
IMHO: Downpicking would be faster/more comfortable. Alternate picking would work if you'd string skip and get back up, if your going in one direction, might as well go with the flow!
Down pick every note, but not a sweep motion. Separate pick motion per string.

Up pick when its reversed.

To speed that up, I found finger strength exercises great for that, for example trilling with each fingers combination. Many excercises can be found online.
I'm currently working on the exact same thing. Downpick it, but try to make sure it sounds about the same as your hammer ons
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For patterns like that i just use Down strokes, if it was string skipping i would use alternate though.