Fixed-gear bike: How to ride backwards?
Bicycles are designed in a way that makes them very stable going forward, at the cost of being unbelievably unstable while riding backwards. The most common fixed-gear geometry (inspired in track racing) doesn't help a lot. As a comparison, BMX is a modality where riders usually ride backwards in half-pipes, but the bike geometry is very different.

What I tried to do (before giving up), and what I have seen people doing, is to try to ride backwards IN CIRCLES, perhaps lowering the saddle a bit to a more relaxed position. I don't think the clipless or toestraps are key here, but a lower gear ratio might be very helpful.

After you perform some circle attempts, you can try to switch sides. Trying to do it very slow aids to gain inside as to which movement you actually have to do, since this movement is very counterintuitive: when you are leaning SLIGHTLY to one side, you have to point the handlebar the opposite way, so that the bike will go under you again.

I could manage to go some 4 meters back when I was trying often, but with almost no control over the direction the bike was going.

Perhaps a good video to start (with interesting links on the sidebar) is this:


                
            

Hope this helps!