Iodine is supplemented in salt in most developed countries. It is the backbone of thyroid hormone---no iodine-->no thyroid hormone. If the thyroid does not have enough iodine then it tries to compensate by generating more machinery to take advantage of any scrap of iodine available. This presents as a thyroid goiter (enlargement of the thyroid gland). They are very common in underdeveloped countries, but can also develop in the absense of iodine deficiency in some people for varying reasons.
Checking iodine levels in a patient would universally result in very, very high levels (well over the level needed to make thyroid hormone) so it is not very useful and does not tell us anything about the ability of the thyroid to make thyroid hormone. It would be like checking the octane level in your gasoline when investigating why your engine in your car isn't working. You know there is gas in the tank and you'd be much better off spending your time looking for problems with the engine than testing the octane level of the gas....