5 weeks is way! too young to be leaving its mum. He would have only just started weaning and would still need mums milk and to learn essential skills off his mum and litter mates. I hope its not cows milk you are giving him because milk is not good for cats. I would say get some kitten milk like lactol to replace mums milk at such a young age. Although he would have been much better left with mum as he could develope behaviour problems later on.
If the lady you got him off is selling them at such a young age then she is being very irresponcible and she is clearly not thinking of the kittens wefare.
If i was you i would take the kitten back to its mum for a few more weeks 3 at most. He will be a much stronger kitten and more able to cope with the outside world.

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Don't worry he is getting all he needs as i'm a pro at rescuing kittens now. He's 6 weeks old today. And she was selling them regardless so im happy I got him than someone who would think less of watching what he's eating etc. And his behaviour should be ok as he has my other lot to keep him in line and he's made friends with roughly half already. Plus both my litters their mum's got bored at around the same time and the ones I kept have grown up fine.
He's a happy little chappie anyway, eating and drinking well, using his tray and toddles off to his bed when he's tired

Oh and he won't be an outdoor kitten, too many bad things happen where I live and I'm right near a busy road