A bullseye type ring around what seems like a bug bite can be a couple things, if there is still a bug attached it may be a tick, otherwise it may not be a bug bite but instead ringworm.
What you might have is Lyme disease. You may have been bitten by a tic. The first sign of Lyme disease is a bulls-eye rash. You should see your doctor.
If your insect bite is from a tick, you may be developing Lyme Disease. If this is the case, you should be evaluated by a physician quickly and started on antibiotics to prevent the long term complications of Lyme.
To be honest that does not sound like an insect bite it sounds like you have ringworm and you need to get in and see the doctor there is cream they can give you to get rid of it because no matter what you do it will not go away on its own.