Rechargeable battery pack for circuit
I have this circuit circuit diagram and I have to somehow integrate this into a mobile phone cover.


No, because that'd be bad: placing conductors around the place where your mobile phone has most of its antennas will make the phone work way worse. Don't do that.


  So to power it, I was thinking of using a 3.7v 600mAH Li-Po battery


Wow, that's a lot of energy. Your microcontroller should run years off that, if designed appropriately.


  I know I will probably need a booster somewhere (to get the voltage from the battery to be 5V)


Clear indication that you should use a different microcontroller: Only very few microcontrollers these days need 5V, and all convert down these 5V to way, way less for their core.

So, dump the gigantic and ancient Atmega for something smaller that can directly run off battery voltage (or lower).


  how do I begin going about making this a rechargeable arduino project? 


So, considering you've posted a wiring diagram instead of a schematic, and forgot to tell us what this thing actually needs to do:

Start by writing down requirements for the finished system. Only after that decide on the components: I kind of doubt you'd be best served with any Arduino-based platform, but even if you are, I don't see any reason to use the enormous Atmega168.

So, as said, start from specifications. Break down your problem into smaller problems, like finding the right components, drawing a schematic (instead of a breadboard wiring diagram – you can't carry a breadboard around, anyway), coming up with a mechanical drawing that shows what needs to go where, converting the schematic into a layout based on your mechanical constraints, and finally building and testing everything.

Notice that you'll really need to learn how to make PCBs (it's not hard) to make a flat, reliable device (breadboard is not only clunky, it's also terribly unreliable when carried around!).

In any way I turn this, really learning how a schematic abstracts your circuit to its connections, and doesn't "mix" them with the physical layout like your diagram does, is paramount here: I have not even checked your circuitry above for "correctness", because it's a lot of work seeing what things are meant to be when all you see is a wiring diagram, but it's easy when you have a proper schematic.

Baby steps – start small.