Routing power and HDMI signal into usb type-c for DIY computer monitor
Well, yes, but not in the way you describe. There are a few things you need to be aware of.


while some USB-C ports have support for external monitors they are either DisplyPort or Thunderbolt, HDMI support comes by way of special (but not unobtainable) chips that can convert DisplayPort to HDMI (these chips are only one-way so check closely which one you have)
while you can theoretically tap into the power of the USB-C PD (Power Delivery) spec, not all computers have full USB PD support in that most can only manage 5V out, while 9V/12V/20V output is usually only seen on USB-C chargers (the PD specs have four official voltages and several current ratings) You'll also need some interface chips so your monitor can tell your PC both which voltage and how much current it needs (NOTE: The PC can always say "NO, that's too much, I can't do that")
Be cautious about Thunderbolt, Thunderbolt only speaks one language, Thunderbolt. If you specifically want TB support, well, tough, 20Gbit PCB design is about as straight forward as a heart transplant... BUT, Thunderbolt controllers have support for the DisplayPort over USB-C protocol (in which case, you're using DisplayPort over a different connector and not using Thunderbolt thus sidestepping the problem).
Be aware, not all USB-C cables have all 4 high-speed differential pairs connected or support higher power loads (USB-C 2.0 only cables exist because... confusion is good right?), even if you had a monitor that natively supported DisplayPort, you'd still need lane switching chips (reversible connectors aren't magic after all) and some chips for negotiating an appropriate amount of power from the PC.


So while it's entirely possible to make a fully fledged USB-C monitor interface adapter, it's probably easier to get an active USB-C to HDMI adapter, rip out the guts and try to hook it into your LCD driver board. Unless your monitor only needs a few watts (then you can boost the default 5V up to whatever you need) you'll probably still have to do some jiggery-pokery to get the 12V you need though (at the end of the day you may have to keep the external power brick after all).