This may help you:
I don't write for TV but a couple of my friends do and according to them, getting an agent and selling a TV series is an extremely difficult and competitive process. Even if you manage to secure an agent and they manage to sell the rights to your TV drama, the chances of it actually getting made are desperately low. The reason being that rights are bought for a set period (perhaps 5 years) and during that time production companies run into untold difficulties until the rights return to the writer and you have to start again.
When I went through a low patch with novel writing, I considered screenwriting instead and they talked me out of it, saying that publishing is far easier (and I thought publishing was hard enough!). That's not to say you shouldn't keep trying! Always follow your dreams!
However, the reason for pointing this out is to say that it's a hard enough business as it is, and if you write something that has the potential to lead a production company into litigation, you're stacking the odds against yourself when the odds are stacked enough as it is.
Does it have to be a real-life star? Couldn't you write your story using your star as an inspiration for a fictional character, making sure you change enough details to make them unrecognisable and the events that happen to them seem unconnected?
Your story could be just as good, just as marketable.
I'm an old broad, so if you're a youngster you won't have heard of this movie, but look at A Star Is Born (I'm a big fan of the Barbra Streisand and Kris Kristofferson version). The screenwriter creates a completely fictional star (albeit reminiscent of any number of real-life rock stars) and watching it, you truly believe he's real. It's handled so well, I don't feel cheated because that character isn't based on a real-life celebrity.
If I were in your shoes, I would remove as many obstacles from my path to success as I possibly could before starting my journey. It's a hard enough journey as it is.
Good luck! Hope we see your show on TV one day!