How do I claim my credit for hosting a high school exchange student?

Have very low expectations.   Unless you are itemizing deductions like mortgage interest, property tax, etc. the credit you get for hosting an exchange student will not affect your tax due or refund.

From the IRS

https://www.irs.gov/publications/p526#en_US_2012_publink1000229664

Expenses Paid for Student Living With You

You may be able to deduct some expenses of having a student live with you. You can deduct qualifying expenses for a foreign or American student who:

1. Lives in your home under a written agreement between you and a qualified organization (defined later) as part of a program of the organization to provide educational opportunities for the student,
   
2. Isn't your relative (defined later) or dependent (also defined later), and
   
3. Is a full-time student in the twelfth or any lower grade at a school in the United States.
   

You can deduct up to $50 a month for each full calendar month the student lives with you. Any month when conditions (1) through (3) are met for 15 or more days counts as a full month.