(I Play bass)

My playing always seems to be too stiff. Check any one of my vids to see what I mean (youtube.com/user/UnforgiveToBlack)

Any tips on how to gradually stop this from happening?

first I would figure out why you feel stiff when you play (specifically).
then I would learn ways to change those problems by adjusting my techniques and/or mental state while playing - or both.
Finally start practicing with your new relaxed methods a lot until they become natural in your approach to the instrument.

Your question is very general, but I'm guessing you play with some excess tension.
If so, you need to analyze where it happens and get rid of it. good luck.
How long have you been playing? May just be an issue of not being comfortable with it. Playing guitar naturally without any stiffness will come with experience. Simple as that.
Well......I have big balls.
When learning a new riff/ solo or something, I'm very stiff, or if around other people!
On my own playing with a scale or riff I know, I'm loose and relaxed.
I think you just need to get more comfortable with it!
Freepower knows his stuffs.

Listen to him.

Oh, and just play the damn thing, they'll mellow out with time.
"If you're looking for me,
you better check under the sea,
because that's where you'll find me..."