How can I drill through concrete blocks to allow drainage from my backyard?
Pictures might change the answer but...

Is the terrain outside the block wall your property?  Is the elevation lower than your backyard?  If you answered yes to both of those questions I'd say strategically and carefully, maybe hire someone, remove blocks or pieces of block. Drilling holes does not sound like adequate drainage but if they were big enough it might work.

I would probably remove a section of brick at the lowest point in the yard and dig a large pit.  Line the pit with some type of filter fabric that will keep fine soil particles from passing through and then almost fill the hole with large gravel. (Here in this part of the US we would use what is called blue stone)  Cover with the same fabric.  Install a porous grate or cover of some sort over it.  You could even cover it with soil and plant grass or something else if you liked.  Figuring out what survives there might take trial and error. This will give the water somewhere to escape to while it slowly drains out of your yard.