Proof NP-Complete for Single machine Job Scheduling Problem specific version
Since this problem is very approachable, you could first try to think about a greedy solution to show this is in the class P, and thus probably not in NP-complete. Once you have done this you could think about a polytime reduction to a problem known to be in NP-complete. Consult Karp's original 21 problems to get a feel for the type of problems you can use to find this reduction. Once you have done this come back with more specific questions :)