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I am trying to order these according to boiling point. I think that B is the highest because it has hydrogen bonding. I am not sure how to differentiate A and C though. They both have about the same molar mass. The only thing I can think is that C has a double bond. Does this double bond make the boiling point of C higher than A because it is stronger than a single bond? Or perhaps the double bond is shorter which decreases branching in C making the boiling point lower than A?

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It's not the double bond in C that is significant here, it's the oxygen. The carbon-oxygen bond is polar and will make C more "electrically sticky" with its neighbors compared to A. Since the $\ce{O-H}$ bond in B is even more polar than the ketone in C, its boiling point will be even higher. Read more about it on this page.

(If you'd like to skip to the relevant section, on Google anyway, search "ketone vs alcohol boiling point".)

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