I have a TSH level of 5.69. Could breastfeeding or pregnancy cause this? Is my level considered high? Could this level affect my menstruation?

Yes. It is very important that you have your TSH checked again. If the level is similar, you must start thyroid hormone medication before you try to become pregnant. This mild elevation in TSH may make it harder to become pregnant. But more worrisome is your thyroid needs to make more hormone during pregnancy and your will become more hypothyroid. This has been shown to be bad for your baby.
Prable thyroid . Deficiency. Your TSH levels suggests thyroxine deficiency. Please consult your doctor for a :"free t4" test and may be replacement therapy. See this site for additional info. http://www.nlm.nih.gov/medlineplus/ency/article/003684.htm.