Assistance required to design a Powerbank
Most Powerbank circuits use ICs and most are surface mount. Learning to solder the larger sizes of SMD ICs is not too hard and well worthwhile.

By looking through circuits that are described as boost converters and which are suitable of using a LiIon cell for input and having a 5V output you should find a number to choose from. Ask further once you have found some possible candidates if you need more advice. 

2 transistor self oscillating converter:

Here is a suggested 3V to 5V boost converter circuit. This is not a marvellous circuit compared to IC based ones.



MintyBoost:

Here is a powerbank kitset for a device known as a MintyBoost - This is usuallyt used as a 2 to 3V input circuit but should work with a LiIon cell input. The circuit is available and you can build it yourself from parts obtained elsewhere if you wish. This version uses an SMD IC but I think earlier versions had a DIP IC option.

NCP3065 / MC34063 boost converters:

Boost converters available in DIP packages are the NCP3065 and MC34063. Both are old design ICs and not as efficient as more modern ICs but still useful and they have a low external parts count and are easy to use. 

NCP3065 data sheet

MC34063 data sheet

Figure 6 in the MC340-63 data sheet shows a boost converter which would form the basis for experimenting with. 

Example circuit below. The values that have been struck out in red were specific to a particular voltage in and out. See data sheet for details on values for eg LiIon in &amp; 5V out.