Cadence

I'm a relative beginner, but I've seen people saying that you have to be careful not to overstride when trying to run faster because you might injure yourself.  That would seem to imply that you have to push harder on each footfall (so you travel further on each "in the air" phase) or increase the number of strides per minute.

 Assuming the first is based on muscle strength and won't really increase unless you do hills or strength training of some other sort, is there a way to get faster by increasing the number of strides per minute?  Is it possible to train this and if so, how?

Shorter , faster strides, simple as that

If you do build up your leg strength then you can increase your stride.

If you only have short legs then increasing your stride rate (cadence) is probably going to work better.

You need to cover more distance with each stride but without over-striding.  Have a look at the "Chi Running" book to understand how to do this.

Short fast strides may give you the illusion of speed but won't take you very far.  Stride length is important and the way to improve it is through push off the ground and overall muscle strength not through reaching out with your forward leg (sounds like you have a good understanding of this already).  You want your feet to land under your centre of gravity - most elite middle distance runners have a stride rate of 180 steps per minute (you can count this with a watch - count right foot-strikes and double the no).  This is a good indicator of whether you're under/overstriding.  It is however true that overstriding is more common than understriding, especially with beginners.

I think  my stride length is probably about right at the moment, which is why I was concerned that I didn't make it any longer.  I'm sure my muscles have got stronger since I started (I have visible thight muscles now), and I'm faster for it, but surely leg strength doesn't keep increasing after a certain point?  Which presumably leaves increasing cadence, but I don't really know how to go about increasing it (although I haven't managed to measure it yet, I'll have a go at the weekend).  Are there any drills that help to increase it?  I had in mind something similar to the army running through tyres thing (but without the tyres).

A drill sprinters use is to run down a steep grass slope - this gets your feet used to moving fast.  However, it is controversial because best thinking nowadays is you should leave cadence alone unless you demonstrably have a problem with over or understriding.  Stride length however can be improved mainly by improving toe-off  or force off the ground.  There are other things you could focus on that might help you more like, for example, reducing upwards movement and ensuring that when you run main force is forward/horizontal.