USB type-c power supply (up to 20V 5A) from DC source
Form the set of keywords "USB", "type-C", "power supply" "up to 20V 5A" and the literal question,


  I want to make a power supply to charge my laptop using this power
  bank. It's not a big deal to make a 5V charger ...[snip], but what
  about higher voltages?


it is obvious the that the question is about implementation of Power Delivery protocol over Type-C CC lines and design of power supply (controllable DC-DC converter) with variable output in accord with Power Delivery handshake protocol.

Answer: there are several semiconductor companies who offer a portfolio of ICs and design guidances on how to build a PD over Type-C.

One is Texas Instruments, see their presentation on USB Developer Days 2017.

The other is ST Micro, see their guide.

Cypress Semiconductors offers a portfolio of reference designs as well.

This is from just the first page of Google search. There are several other companies like Maxim, Fairchild, NXP, ON Semi, Renesas/NEC, Rohm, and maybe many more Chinese sources with similar products.