You may run into some compatibility issues with this frame, and should be aware of potential issues, or sure of the correct size/standard, before you buy certain parts. Just knowing that it is a Peugeot isn't enough, as they were made over several decades on different continents with different standards in different places. The wheel size will be either 27" or 700c, probably 27" if it was made for the US. That means most modern road wheels (700c) will be a bit too small (8mm), so you would either have to have a "long reach" brake to reach the braking surface or make sure you have 27" wheels. The axle spacing (wheel width) might be a bit different too, which can usually be corrected with some fiddling with cone wrenches and washers, or just calling it good it if the difference is not more than a few mm's.
As for the bottom bracket, a lot of peugeots have the less common "french thread" bottom bracket shell (there are a few different standards, "english" being the most common), meaning most new bottom brackets wouldn't fit. If you can spin the cranks and there is not a lot of resistance, and there is no noticeable play or rocking in the axle, then you probably don't need to worry about it (I would be sure to tighten down the bolts holding the crank arms on though, good practice).
I would do some research about what size wheels and bottom bracket that bike has, which you can probably just find with the serial number under the bottom bracket.
It looks like you really just need wheels! A back brake wouldn't hurt, also ...