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Today, the company is amping up its search capabilities again, this time adding "caption search" to all shows that support closed captioning. Just search for a particular word or phrase in the captions tab on a supported show and you'll get back results that you can immediately access.
The captions search results show up with thumbnails, so you can visually select the part of the clip you want to watch (which is helpful for more generic searches). Hulu has also implemented a "Heat Map" that shows the portions of an episode that contain the words or phrases you are searching for. You can click on the blue lines in the map to go straight to those sections or more generally visualize your searches' placement in the clip.