What are his chances of staying if he is currently in a deportation process but is eligible for deferred action?

If your boyfriend is already in removal proceedings, he should immediately retain an attorney, if not already. The attorney can ask the DHS for exercise of prosecutorial discretion PD and it is likely that DHS considers his case not worthy of DACA or any other discretion. If you are a U.S. citizen, you should consider marriage to him that will form the basis of an adjustment of status green card. Even if you already have an attorney, I believe your boyfriend will benefit from a second legal opinion regarding his case.

If he is certain that he is eligible, he should file his I-821D request for deferred action for childhood arrivals with USCIS immediately. As soon as he gets the receipt for his case from USCIS, he can file a motion with the immigration judge along with a copy of the receipt, his DACA request form, and supporting documents to ask the judge for administrative closure on the basis that he is eligible for DACA. The motion should be filed beforehand but may also be submitted in court if necessary.