I think that you will have to go about deconstructing their relationship. Perhaps, a starting point would be to suggest that of all the aspects of Winston's attempts at resistance, it is his relationship with Julia that represents the deepest and most provocative action. While he might have thought and done different things to voice his discontent with Big Brother and the government of Oceania, the relationship with Julia is probably the largest step taken, the greatest risk assumed, and, by extension, the largest amount of repercussions felt. Naturally, the fact that another person is involved with his resistance is one reason why there is danger present and another reason would be that the act of sex is strictly forbidden by the government. Their relationship is predicated upon this disobedience. I think that this might be a starting point in constructing an argument that the relationship ends up being the straw that broke the camel's back, in a manner of speaking.
What I would do is first to think about the things that happen to Winston. What, specifically, causes him to get in trouble? When I think about that, I would think about his relationship with Julia, of course. It is the clearest thing that he does that is against the law. However, he does also keep his diary in which he writes things like "Down with Big Brother." Remember that he had already started doing stuff like that before he even knew Julia's name.
Second, I would ask if the relationship with Julia is what caused him to get caught. Did he get careless because he was in love? Or would he have gotten caught no matter what?
Personally, I think that it is not the relationship that dooms him. I think he is doomed from the start. I think that it is his desire to have a relationship -- his unhappiness with society -- that actually dooms him; not the relationship itself.