How should I address this window leak?

I wouldn't disassemble the sliding part unless it seemed like that's where it was leaking in. I'd probably start with general inspections all around, pulling the rubber bit out of the channel--the one that's not seating all the way--and inspecting the path to the weep hole, as well as checking that the seals along and at the ends of the slider part are basically whole and not leaving a gap. The general theory is not that it's impossible for water to get past the seals etc., but rather that the quantity is quite limited and the channel lets it flow to the weep holes where it dribbles out harmlessly rather than collecting enough to get things wet inside. Since the latter is happening, either the water can't get to the weep holes properly or there's so much collecting in the channel that it is overwhelming the drainage system. I'd guess the former is more likely the cause of the trouble.

Obviously, it's good to have the slider part seal as well as it can when it's closed, not just for the sake of preventing water from entering but also to prevent drafts, bugs, and the like.