As others have indicated, your doctor is probably clueless wrt Iodine supplementation. He/she would probably try to dissuade you from taking Iodine beyond the RDA. Having said this, if you have not been to a doctor in a while I think it is a good idea to have the complete suite of blood tests, including thyroid tests, that most doctors would give adults as part of a major physical exam. You wouldn't want to start Iodine supplementation if, for example, your kidneys are having issues. If you have insurance these blood tests shouldn't be too expensive. ... Again, this is just my opinion.
As with most alternative treatments, iodine supplementation is frequently done w/o any oversight of a physician. Physicians are usually not too keen on dietary/alternative approaches to resolving chronic medical issues. They are trained to diagnose and medicate. And so going alone with iodine supplementation means you are taking 100% responsibility. I think this is okay provided you do your research, are disciplined and take iodine supplementation *very seriously*.
PS - I am big believer in going with dietary solutions first before looking to supplementation, including iodine supplementation. Depending on your health complaints you might want to examine your diet/lifestyle first before starting up on iodine.