"That" refers to sex and/or drugs
Billy Costigan and Madolyn had an intense, surreptitious sexual relationship. Because Billy could not really be himself, tell her the truth, Madolyn came to believe he was just play acting his vulnerability to get drugs from her at first and later get her into bed.
Madolyn: What do you expect coming in here?
Billy Costigan: I have to come here.
Madolyn: I know you have to come here, but now that you're here, what
do you want?
Billy Costigan: You want the truth? Valium.
Their connection had gone sour by the time Billy gave her the package, so his approach to her could not help but be awkward, especially seeing as how he likely had developed genuine feelings for her and knew he had blown his opportunity. Getting her to help him was crucial, so he wanted to make clear that what he was asking her to do was beyond their personal attachment. It's also likely that Billy hopes the information contained can save Madolyn from getting ensnared in Colin's evil life.