While you are free to marry whomever you wish, this individual will not be able to acquire any legal status through you anytime soon, at least not until you become a US citizen, which is at least 5-6 years away. Even then, if that individual entered the country illegally, your being a citizen will not help him any. Now that you've been finally granted asylum and will start a brand new life in the US, do you really want to saddle yourself with such a burden to which there is no legal solution (nor do we realistically expect there'll ever be any solution) ?
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
Your approval as an asylee will not assist your boyfriend at all even if you married him as it is too late to do so. If he entered the U.S. with inspection by customs officials, as a US citizen you would be able to petition for him, but that is many years later. If he were not inspected, he and you will have to live with his status being in limbo potentially forever. Consider your decisions carefully!
I need more information related to your boyfriend's legal status. Did he enter on a valid visa and then overstay, or did he enter the country without inspection? Also, you will need to wait a year until you get your green card.