Can I claim my godson?

I assume that the child is not related to you.  A "godchild" relationship is not considered related. 

He/she may be a “qualifying relative” dependent if:

…He/she is not filing a joint return with anyone else

.…He/she does not file a tax return claiming someone else as a dependent

…He/she is no one else's qualifying child

…He/she lived in your home as a member of your household for the entire year.

…His/her gross income was $4,000 or less (2015).  Social Security benefits that are not taxable do not count against this limit.

…You provided more than 1/2 his/her support for the year.

…Your living relationship did not violate local law.

…He/she is a U.S. citizen, U.S. resident alien, U.S. national, or a resident of Canada or Mexico, for some part of the year.