I would either till them under or put a black tarp over them for a few days to kill them if you don't till. No sense in loosing that green organic matter by using Round up or something similar.
Good ole fashion hoe and rake. I prefer not to till an older area and a rake usually pulls weeds out easily. If a rake can't handle it then the hoe comes out and get em.
I usually opt to smother them with a thick layer of mulch. I usually add that thick layer of mulch in the fall, so the weeds are dead and (mostly) decomposed by the time spring rolls around. If you mulch year round, you'll keep the weed growth to a minimum.
Though I'm new to peppers, I use a coconut and pine oil spray on the rest of my vege garden when I need to clear weeds - it pops the cells of the plants (from my understanding) and so renders them 'inert' while still retaining their inherent compostability.
The other thing I often do is, when I know there's going to be a couple of days with lowish wind, put a couple of layers of newspaper over the weeds, heavily wet it and leave it, then dig the whole lot in (if in a small area or where I don't want to risk overspray).
I'm with compmodder. A layer of 4-5" of wood chips keeps weeds from popping up.
If any get through that the ground is usually so soft they pull right up easily. Then they get thrown on top to dry in the sun and feed the plants I want there.