Efficient self study plan
I'm not super familiar with Bishop, but it looks like Durrett, which deals with measure-theoretic probability, would be overkill.  Plus you would probably have to add real analysis to your prerequisites.  I would just work through the first 4-5 Casella chapters on probability and random variables. 

You probably also don't need to go through the whole linear algebra book to get started.  Just make sure you understand the basic matrix operations well enough to deal with multivariate calculus (ex. taking multivariate derivatives and computing multivariate integrals).

The second half of Casella would likely be useful, but I suspect not strictly necessary.

I think if you can handle those things you'll have most of Bishop covered; you can always pick up more as needed.