I had Tricare Young Adult Medical Insurance which I paid per month after taxes because it was not taken out of my fathers account. How do I claim that?

In terms of your medical expenses - you can claim your medical insurance premiums as medical expenses. However, if you are not itemizing (have mortgage interest, property tax, charitable donations), the expenses will not make a difference on your tax return.

In terms of health insurance coverage - Tricare is not considered Obamacare.  When going through the Health Insurance section, all you have to do is indicate that you have health insurance coverage all year (if applicable) and that you were not enrolled in other plans (such us Obamacare of Healthcare.gov plan).