To firearm hunt you would need a firearms hunting license. If you have never had one then you would need to take the Hunter Safety Course first.

For archery, you would need an archery permit. Same deal, you would need to take the archery course before obtaining your permit.

Once you have your permit/hunting license, you would be permitted to hunt coyotes during their regulated season (Jan 1 - April 27
May 31 - Oct 1, Oct 16 - Nov 15,2010)

If you want to keep the pelt, you would need to have it tagged by DEP. If you don't want to keep it I would suggest not wasting it, rather, post something here on this site, there are enough trappers on CTF that I'm sure one would probably take the carcass for the pelt.