I assume by single speed - you meant a single freewheel as supposed to fixed gear or existing 5 speed freewheel. To convert to fixed gear you will have no choice by replace the rear hub. Of course, you can also keep the existing 5-speed freewheel, and use only one of sprockets - it wouldn't be too hard to find the right one to align the crank chain ring. But I personally hate the look and do not recommend it.
Assuming you are planning to use a one speed freewheel, there are a couple of options. The easiest which is what I usually do is to replace the drive train as well as the wheel set. (I usually do both wheels but it is the rear that matters). This is not the cheapest way, but it is hassle free, you got to choose a crank/bottom bracket that is made for single speed, and possibly a flip flop of fixed/single rear wheel, plus you also get to choose the bolt down hub than the quick release!
If the budget is limited, you might have to choose what to keep. Technically, you can opt to keep both hub and crank. But it gets pretty tricky to align the drive train. The key for the single speed bike is to keep the drive train perfect straight. (Yes I meant perfect straight). It looks like you got a pretty good crank, I'd try to keep that. You might have a better luck if you know someone that has access to different length bottom brackets. It is not too hard to find different length of the old square bottom brackets.
Finally, if you really have to use the same rear wheel (it would be the last thing I'd consider to keep), it would be a lot of work. You will need to find the right "center" by adjusting the spacers and the crank, then "re-center" the rim... I have seen people doing it, but I am too lazy to even give a try.
Good luck and keep us posted.
