I think this really depends on the company's culture and size. From my experience, having only worked in smaller teams, I'd say trying to partake in management duties proactively has probably been most successful for colleagues who wanted to lead.

So when your boss or supervisor has a meeting about your product, ask if you can join. If you have a well thought-out idea on how to improve things, like introducing better processes, fixing recurring issues, introducing better tools or something like that, talk about it. Being visible as someone who genuinely cares about the success of your team, product and company is, in my experience, probably the most important thing.

Just make sure this is actually what you want. Depending on the company, you might end up doing very little programming and lots of spreadsheets and misery instead. Find out what's keeping your current team lead busy and ask yourself if that's really what you want to do.