What's wrong with my nikon d5000?

My battery charger blinks and blinks for days but never stays lit to notify me with full battery charged. Then I put the battery into the camera and my camera won't turn on. Is it the battery or the charger that I need to replace?

It may be that your Li-ion battery has become damaged.

How can this happen?

Never let the battery be left fully discharged.
After about 400 recharges, the battery will eventually become less viable and eventually not charge again.

The Nikon Li-ion battery charger will NOT overcharge or overheat your battery. If for some reason you are using a generic charger, that too can cause problems.

I have had Nikon Li-ion batteries and chargers for over 10 years and all still work just fine.

Call Nikon and tell them what problems you are having and they may have a way to help you troubleshoot which is the problem, the charger or battery. 800. NIKON-UX

Replace both to be absolutely sure.

I'd try replacing the battery first.