You can take the airport buses (9am to 8.45pm) to go to pettah, and take a bus (not sure about timings) to Jaffna. 1700 LKR in Kangaroo cab service. About 2500 otherwise (plus toll fee about 300). A mid range sedan. Vans will cost a bit more.
You would be quoted ridiculous prices if you take a taxi from Colombo to Jaffna. You could even fly for a fraction of that cost if the dates work.
There is a much comfortable high seater set of buses running from Colombo to Jaffna (via kurunagala I believe).