Budgies bite for several reasons, and it's not always a sign of illness. Often it's territorial aggression - he's simply protecting his cage and has gotten used to it as his own space. There can also be hormonal arousal, especially if the bird has reached sexual maturity (around six months to a year). Sometimes bites are the budgie's way of getting your attention or showing boundaries if he feels dominant in the house.
You should look into his living conditions, but biting itself doesn't always indicate health problems. Check whether the bird has enough space in the cage, whether he gets a normal day-night cycle (budgies need about 10-12 hours of light), and whether he has toys and variety in his diet. If otherwise the budgie is active with a good appetite and healthy plumage, it's most likely a behavior issue.
To discourage biting, don't just stick your fingers in the cage - let him get used to hand movements from a distance. Offer treats on a stick or through the bars instead of with your fingers. When you pick him up, do it confidently and calmly, without sudden movements. If he bites - don't yell or jerk away, just gently put him back. Over time, if you show him you're not a threat or prey and give him the freedom to choose (not forcing him), the aggression usually goes away.