Am I obligated to give my relative a sum from my salary monthly, after I intended to do so but did not explicitly promise it, and would my salary become unlawful if I do not?

You are not obligated to give the amount you intended to this person, because a promise is not binding and does not have to be fulfilled according to the majority of scholars; rather, it is supererogatory. Similarly, the intention to do good does not make it obligatory upon a person.