I'm basing this off of the assumption that "he" is the one doing the reaching. He climbed up on the chair reaching for the cup: While most people would probably interpret this correctly, there could be some confusion. The positioning of "reaching for the cup" implies that the chair is doing the reaching. This doesn't make as much sense as if "he" did it, so most people would probably just assume that he was doing the reaching anyway. A better sentence structure would be to say "Reaching for the cup, he climbed on the chair." This makes it more clear that he is the one doing the reaching. This is something that plenty of native speakers say wrong too though, so while technically incorrect, you could probably get away with your original sentence structure. He climbed up on the chair to reach for the cup: This is perfectly clear. "To reach for the cup" is there to explain why he climbed on the chair.