I have a headcold and a constant cough. I am taking an antibiotic but it doesn't seem to be working. What can I do next?

Rest and wait. Probably viral then and sometimes it takes awhile to clear. Otc meds can help symptoms, but won't make it go away quicker. Stay in contact with your doc to make sure you don't have a pneumonia.
Viral? Your symptoms may be due to a viral infection and if this is the case, antibiotics will be ineffective. The infection will likely run it's course over a few days to a week. If your symptoms are worsening, see your doctor again so you can be re-evaluated.
Chronic. If it has only been a few days, you can wait and see whether the cough goes away. A lot of times antibiotics won't help a cough right away, or it won't help at all if the cough is caused by a virus. Most viral illnesses get better in a few days anyway. If the cough is persistent for longer than 10-14 days then it would be important to see a doctor to find out what is causing it.