What could cause a swelling at the back of my head at the base of my skull, and painful lump under my jaw?

Swelling: Lot's of things, including conditions that are serious. I suggest you see an otolaryngologist (ent).

Likely an infection: Sounds like you have an infection with a lymph node enlargement in the neck secondary to that. See your doctor, you will likely need antibiotics, anti-inflammatories and possibly need referral for surgery.

Most lumps, benign: Don't start worrying. The majority of small bumps on your head would be benign (not cancer). Lumps that can be tender are cysts, lymph nodes and less likely lipomas. Still, hard to evaluate it over the net. See your primary md who can take a good history and examen you. Given your presentation it sounds more like lymph nodes. You want to make sure its nothing scary