Ok, it's great that he is acting normal and eating as sometimes kitties will feel sorry for themselves after an injury, stop eating and become dehydrated (this is where trouble can start!). Do monitor his eating and drinking to ensure it remains normal.
It's also very reassuring that he can put some weight on it, as most typically with a break or dislocation they will be unable to bear any weight.
It is worth checking his claws/toes and rubbing your finger up and down the leg to ensure there is no cause of the limp e.g. a thorn or a scab.
While without examining little Charlie it is difficult to say for sure, from what you have told me I am most suspicious that this is a muscle strain/sprain. Young kittens get these all of the time when they are jumping/running and finding their feet. The majority of injuries will gradually improve with rest and time and will sort themselves out with no treatment in about 24 hours.
I would encourage rest (as much rest as is possible in an 8 week old kitten!) by avoiding ANY jumping/stairs/running etc. Try to encourage Charlie to lie down and stay still as much as you can, as this will speed up the healing.
You may also wish to make a small heat pack from a sock and uncooked rice. Pop the rice in the sock and heat in the microwave until warm. Place on the leg and leave on for 5 minutes to ease any discomfort or swelling. Do not force the issue if Charlie doesn't like it. If he tolerates it, this can be done a few times a day
Please DO NOT give Charlie any human pain medicine as this can be HIGHLY toxic.
You should notice when he wakes up tomorrow his limp is much improved and it should continue to improve each day with rest. If it is getting worse at all, or not improved after 24 hours, or if Charlie seems unhappy in himself, a trip to the vet would be in order.
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