I am looking to fish some Cat 6 network cables from the basement to the attic.
Initially, I was going to attempt using my bedroom's cold air return vent, but it appears that it's not a straight shot to the basement. There is a 90 degree turn somewhere, and I don't know how to use my 100ft. fish tool (or any other type of tool, for that matter) through it.
That is what I'm referring to the cold air return vent:
How can I achieve this before I find another solution?
Your "intake vent" is better referred to as a "return air vent in the stud wall cavity". The reason there appears to be a 90 degree turn in it is because there is! Behind the drywall, at the base of the wall, there is a hole where the floor would be, this transitions into a metal duct which travels back to your HVAC unit, just like you see if you open up your floor registers; the air is just moving the other direction.
I would not drill a hole for your Cat6 cable in the attic OR the basement in this stud cavity for this reason. Rather, pick an interior wall stud cavity where you have basement access (try the utility room) and there aren't preexisting electrical runs. Drill a hole from the attic and basement (alignment doesn't need to be perfect, just hit the same cavity). Use a paperclip & string from the top, and magnet from the bottom, to fish a line between the two. Then use that line to pull your Cat6 cable through.
Retrofitting low voltage wire through a finished structure can be a challenging brain-teaser. When you mention fishing through an "intake vent" are you referring to the ducting of a central HVAC system? If so you'll need to use plenum-rated cable, and be aware that depending on air flow through the duct, you might get noise (whistling, vibrating, etc).
Here's a strategy for fishing around a single corner. I've only used it for fishing through walls, ceilings, and floors, but I don't see why it couldn't work inside ducts too.
Get a set of "fish stick," "glow rod," or whatever other name you might find them called by. They're semi-flexible rods often made of fiberglass. Depending on the distance you need to cover, you may need more than one set. Here's an example of one sold under the Greenlee name:
Also get a USB endoscope camera. They're available in your preferred online marketplace with a variety of cable lengths, and some offer WiFi.
Tie a loop in the end of some string (mason's line etc) and fix it loosely to the end of one fish rod. Mount the hook accessory on the end of another fish rod. Fix the endoscope securely to the hook end of the other fish rod, positioned so the hook can be seen clearly through the camera.
Choose a pair of access points into the structure which provide straight paths that converge at the subject corner. Use the one rod to feed the string in from the access where the wire will be pulled to; feed the rod with the hook from the access where the wire will originate. While watching the picture from the endoscope camera, feed both rods until they meet. Use the rod with the hook to grab the loop in the string, then carefully pull it back. Now you've got string around the corner; use it to pull the cable through.
