Sounds like eye pinning to me, but I don't have any experience w/budgies.
http://www.avianweb.com/communicatewithyourbirds.htmlParakeet, does your budgie's beak hitting your finger seem intentional? If so, pairing that w/the eye pinning would lead me to think your budgie is exhibiting aggression, although Lord knows I'm no expert. If it is aggression, I realize your budgie isn't hurting you, but you don't want to purposely irritate your birdie.
How familiar are you w/bird body language? Barbara Heidenreich has a DVD that provides the basics, & it's helpful because it's not just a verbal description - you can actually see birds exhibiting the behaviors people talk about.
http://www.goodbirdinc.com/parrot-store-dvds.html