Good exercise for posterior knee injury prevention?

It's hard to tell without really knowing where these problems come from. "Posterior knee" could really be anything from hamstrings to calves to IT, or, rarely, actually the posterior cruciate ligament.

Strength training or stretching for injury prevention is controversial. It's hard to prove that something prevents injury, as opposed to rehab, where you're healing an existing injury and making it less likely to recur. But it's hard to tell without knowing what's really wrong. You might get recommended an exercise that exacerbates what's going on.

You might even need orthotics. You might need to warm up better before the hard runs. You might need more or a different kind of training to build up to these hard runs.

I really wish I could be more helpful, but these are some of the questions/issues that you might want to look into in order to address why you're having these problems.

Best of luck in figuring it out! 

It has been my experience that achilles pain as you describe is from calf tightness rather than a lack of strength.

I use "the stick" after speed sessions and long runs on the calf muscles and this seems to really help.