Wouldn't use bleach!!
Get some of those antibacterial tablets that you throw in to clean out drinking bottles/camel backs. We use them at work. Just be prepared to rinse it about 100 times afterwards, and only use the recommended amount. Sure cleans it good, but getting rid of the chlorine taste afterwards takes a LOT of rinsing.
Edit - the ones we use are called Milton Anti-bacterial Tablets.
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