Cat likes to grab me with her claws out

by Lorry
(London)

I adopted a cat two days ago. She has settled in very well already, eating, drinking, using the litter box all fine. She seems quite happy, rolling around on the sofa and sleeping, exploring the house, etc.

Sometimes when I'm sat on the sofa next to her she will reach out her paws to me very slowly but she extends her claws and tries to hook into my clothes as if to grab me.

She seems to stretch out her paws a lot and keeps deliberately catching them on things and yanking them out.

I just don't understand why she reaches for me. Sometimes she rolls on her back and purs so I rub her face but then she will start extending her claws/paws towards me again.

I don't mean to flinch and pull my hand away too fast but it hurts! And puts holes on my clothes!
Is she doing it because she wants me to go away?

I have tried playing with her with a toy on the end of a string and she will try to grab it (without getting up) but when she gets it she just holds it and won't let go.

Any ideas why she keeps grabbing me? I want to give her attention if that's what she wants but she's hurting me and the way she rips her claws out on my clothes she must be hurting herself?