When I turn the ignition off the car starts making a buzz and the lights blink off and on. Any ideas ?

 

Alarm system.
Do you have a remote fob for it?
My '93 (4 litre) has a factory fit alarm system but I have never had a fob for it.
The only time anything has ever happened was when the car battery was on it's last legs and the the little red light on the ski slope came on (a bit like a dying Terminator).
My V12 Cabbie has an after market system fitted (again, no fob [note to self-buy better!]) and occasionally it makes all sorts of strange noises but anybody can steal the car if they wish to do so.
If the car still lets you in and lets you drive it then ignore it.
It's an XJS, you will have absolutely shed loads more stuff to worry about very soon!

 

No remote. The beeping sound it coming from a little black box under the steering column. It will definetly bring the battery down if I don't fix it though. I was also reading would this have anything to do with the radio not working ? I understand there is a radio shutdown something with this car.

I also need tho replace the heater core int he car which I read is no fun job.