This is an interesting question to consider. If we look at the text carefully, we can see that the first time Katniss mentions the prospect of winning comes after Rue is dying from the spear wound in Chapter 18. Katniss has of course by this stage blown up the food at the starting point, which is a major accomplishment. In response to Rue's command that she "has to win" the Hunger Games, note how Katniss responds:
I'm going to. Going to win for both of us now," I promise.
This is perhaps the first recognition that Katniss shows of the possibility of her victory, and also this comment shows great determination. At this point, of course, the contenders have been thinned down significantly, so Katniss does have a real chance. After this stage in the book, as others die and Katniss remains, obviously her chances of winning become greater and greater.