Making your own version of a similar site is a months-long process and requires a lot of experience in programming (Python, JS, PHP), database, system administration, etc. TM had the advantage of building off the old CHM code and data I had the advantage of some pretty extensive experience with this stuff when I developed my educational tools based off CHM. Even so it was still a major undertaking, and far from complete even now.
I assume instead you'd like to create a site that leverages TM's existing code base, tailored for bike routes. You're more than welcome to do so, but you'll need a Unix-like system (MacOS, Linux, FreeBSD definitely work, not sure about Windows with a command-line extension) to run site updates, and a web server with Apache, MySQL, and shell access. Everything I use to keep TM running is in GitHub. I'd recommend trying to get your site up as a clone of TM first then replace the highway-based data and list files with corresponding bike-based ones.
From the perspective of TM, I'd hope you'd do anything like this as a fork of the TM GitHub repositories so any new code you develop (maybe new mapping or statistics capabilities) that might be useful to the mainline TM could easily be contributed back.
