I (co-writing) with two others on here have had a couple of TV credits. I started writing in 2005 and it took till 2014 (last day of the year!) to get the first one. And it's not exactly rolling in opps since then, it must be said.
The path we took is basically write for open submissions shows, get a lot of credits, get together with other writers and collaborate on a podcast, get friendly with talent, talent introduces you to other radio producers, other radio producers asks you to submit to shows, impress producers with more radio credits, one producer passes your name on to other (TV) producer, write an absolute shitload of material for TV producer, they use a line of it. You're a TV writer!
By the way, it is not 'linear', despite how that path above looks. Every bit of that path has its own branches that you follow up on: ie talent also introduces you to other talent for gag writing and their contacts etc etc. The 'lots of credits' bit at the start, means you already have lots of producers contacts to contact as you finish your sitcom that you may never finish to an acceptable standard.
Traditionally though, the route is: get lots of (non-comm) radio credits, write a f**king good sitcom (this bit is *much* harder than you think it is), approach the producers of those shows (as they will remember your name) asking them to read your sitcom, they are blown away, invite you in to develop your sitcom. Have three or four series of sitcom on Radio 4. BBC think it's a good idea to develop a pilot for TV. Then a series. Then America want to make a pilot version of it. If it's successful, America go to series (at which point, if you haven't already, you have to relinquish control.) BTW this is the only point at which any of the venture makes any significant money.
Jon Plowman said at one of the comedy writer conferences that you will only make money after about the third or fourth series of your sitcom (I assume he was talking about radio rather than TV).
Bill Dare has said, at the last BCG comedy conference, he's never known anyone with a 'career' making it in comedy. They've always had 'jobs' (ie waiters, bar staff, stuff of a transient nature that you don't have to put 100% effort into). (The way we personally collaborate is supposed to offset this small problem!)
Facetiousness aside, the most important things I've learnt are:
-- (David Isaac's advice of) follow up on every opportunity, however small it seems. EVERYTHING! And do lots of stuff for free.
-- collaborate. It is much, *much* easier collaborating with people to get stuff made. Also, your sketches get edited by the other writers and are, by definition, better, funnier and more appealing; and you'll have (number of people)x as much stuff to submit! A corollary to that is, collaborate with people who have a slightly different sense of humour to yourself, so your stuff is not just funny to people with your sense of humour.
-- find a writer with a 'can-do' attitude. They'll soon realise they're a producer, not a writer!
-- try get stuff made. Loads of great actors out there trying to get work and willing to do stuff for free (especially when they realise you are doing it for free too!). Approach them after shows (like News Revue, with its constantly rotating cast) and get to know them, tell them you've written stuff and suggest they'd be great for one of the roles, meet up and record it. Edit it in Audible, put it on the web. Repeat, do it better. Ad infinitum. You'll get better, they'll get better, someone will spot it. If they don't, the public will and advertisers will and you might make enough off advertising to fund future projects. Don't give up.
-- it is not in the least bit profitable, or cost-effective.
Hope that is in some way helpful.