How can I claim my tuition costs if I did not receive a 1098-T form?

It depends on why you didn't receive a 1098-T.  After telling TT that you didn't get one, you will be asked if you qualify for an exception.  Answering yes gets you into the education expenses interview. 

Form 1098-T requirement.

To be eligible to claim the tuition
and fees deduction, American opportunity credit, or the lifetime
earning credit, the law requires a taxpayer (or a dependent) to
have received a Form 1098-T from an eligible educational
institution.
However, a taxpayer may claim one of these education
benefits if the student doesn’t receive a Form 1098-T because
the student’s educational institution isn’t required to send a
Form 1098-T to the student under existing rules (for example, if
the student is a nonresident alien, has qualified education
expenses paid entirely with scholarships, or has qualified
education expenses paid under a formal billing arrangement). If a
student’s educational institution isn’t required to provide a Form
1098-T to the student, a taxpayer may claim one of these
education benefits without a Form 1098-T if the taxpayer
otherwise qualifies, can demonstrate that the taxpayer (or a
dependent) was enrolled at an eligible educational institution,
and can substantiate the payment of qualified tuition and related
expenses.