You can only claim someone that's not related to you if they lived with you the whole year and made less than 4,000 total income for the year.
In addition to the requirements that VolvoGirl stated, you must have provided more than half of the person's total support for the year. Tuition alone might not be more than half of the person's total support. Support includes housing, food, clothing, transportation, and all other living expenses.
The $4,000 income limit is for 2015. If you are working on your 2014 taxes, the income limit is $3,950.
If you mean "how can I claim the tuition payments" as a deduction on your return then the answer is you cannot.
The student (or whoever claims the student on their return) will be able to claim the education credit using those amounts that you paid on their behalf (since it is technically a gift to the student).
If you paid the school directly then the amount paid is not relevant however if you gave the student more than $14K in one tax year then you may need to file a gift tax return. See the rules here: https://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Filing-Estate-and-Gift-Tax-Returns
According to IRS Publication 970, you can claim a credit or deduction for education expenses that you pay for "yourself, your spouse, or a dependent for whom you claim an exemption on your tax return." If chrispsupreme can claim the non-relative as a dependent, and he/she paid the non-relative's tuition, then chrispsupreme can claim a credit or deduction for the tuition. That's why the requirements for claiming the non-relative as a dependent are relevant. There is no requirement that the student be a relative.