1) Apply shine drops liberally to your hair before you put the cap on. The hair will pull through a lot easier if it's slippery.
2) If your hair is long, forget the cap altogether because it's just going to get all tangled up...or find a beauty supply that sells these:
http://www.totalbeautysupplies.co.nz/consumables/sundry-consumables/shrinkable-processing-cap-sunglitz-pkt-20.cfmThose are heat-shrink frosting caps from Sunglitz and hair slides through them much easier than with the traditional caps. You might need to buy a super small-gauge crochet needle while you're at it but one should come with the store-bought frosting kit.
3) Don't pull hair from all over your head. Just concentrate on pulling pieces of hair from the frame of your face and the top/crown of your head.
4) Consider not using a frosting kit at all and instead opt for buying a frosting cap/needle and applying a lighter shade of permanent hair color. Bleach is the hardest on hair and it usually strips all the color out and you might end up looking gray as opposed to blonder. EXAMPLE: On dark blond hair, apply a light beige/blonde permanent hair color via a frosting cap.