Why doesn't my cat let me pet him?

you need to read into his behaviour a little more. He might just be playing as he is young and being feisty - but does he seem to be hissing or defensive? look at his body language and tail. google cat body language and see if he is stressed or not. If he is just playing, make sure to discipline that behaviour or he might bite when older. A simple tap to the nose and a stern voice should do it.

Some cats are overly stimulated by petting and that means they claw/bite... it is a young cat so often more likely to get over stimulated, instead of petting play with the cat, such as chase/catch games with a 'fishing rod' type toy or dragging a sting get the cat tired and let it calmed down before you try to pet the cat....and use something like a gloved hand or one of those car washing mitts that way if it does start to become over stimulated ( because you are doing too much too soon) you can just pull your hand out as it grabs the glove/mitt and you don't get scrached or bitten, just the glove/mitt does

Some cats are a little moody about how and when they like to be touched or petted.

I have a cat that loves to have her head petted and her neck and face scratched but the minute you touch or pet anything below the shoulder blades she turns into the meanest cat - all growls and scratches.

When you have a cat that doesn't like something - avoid doing that thing - or only do it for a very short time -stop before the point that they usually become angry over it.  Some cats will get over not liking to be touched - other cats won't.  Find the locations that your cat likes and stick to that to have a completely happy cat.  With my cat - she does sometimes allow her back to be petted - but usually only a few times and you can usually tell when her mood is about to change.  She is older now and never completely adjusted to liking her back touched - but if you pet just her head - she will lay in a lap and purr for hours.