Earlier quoted context omitted.
If you like dealing with clients (and it sounds like you may be patient enough to be good at doing this), you could try your hand at being a project manager and managing several development contractors for the projects you get from your clients. I built a web development agency like this several years ago. Basically, you start with the same clients that have been hiring you, then tell them you can assemble a team of…
I think I'd like to be a project manager, but honestly I'd be happy doing it at my day job. Any suggestions for how to show people that you can do that kind of job?
There's a general focus in that book about building your personal brand in an organization so you basically have leverage to do what you want.
Beyond that, the short answer is to assume the role. Take on a PM-like presence amongst your team if you haven't already - ie. help any PMs you're currently working with on small tasks, ease their burden where you can - then have a candid discussion with your supervisor at some point about how you can see yourself in such a role.