This is a tougher question to answer than ones about private equity, because the hedge fund world is so broad. However, most investment bankers that move into hedge funds, wind up doing either fundamental type investing or event driven investing (merger/risk arbitrage or distressed). For analyst positions at fundamental based hedge funds, you will need to demonstrate excellent technical skills, especially valuation and modeling. Moreover, you will need to show that you not only have a strong interest in investing, but that you have the ability to make or recommend investment decisions based on your analysis. This is a key difference between the buy-side (e.g. hedge funds) and the sell-side (e.g. banking). In banking, the goal is do analysis that is accurate, with assumptions that you can back up. In investing the goal is to be right. You will need to have the confidence to act on your analysis, not just format it. Note also that many hedge funds, in the interview process, will ask you to either do an analysis of a position or recommend an investment.
Those interested in merger/risk arbitrage or distressed investing are going to need to demonstrate strong understanding of the M&A process or the restructuring/bankruptcy process, respectively.