As an asylum seeker you can get married. You need to contact your local registry office (Standesamt) and request this. They will need at least the following documents from both of you: passport, birth certificate, certificate of eligibility to marry.
If you are married and your asylum case is negative, you can apply for a visa with the French embassy to go and live there as a family member of a French citizen. I do not know, however, what exactly the requirements for this are from the French authorities.