Every time I charge my battery, when I start my camera, it asks me to input the date and time. Why? It's very annoying.
The Nikon D5100 uses a rechargeable internal clock battery. It should have enough charge to give you 3 months to charge the main battery. A main battery that is almost completely empty, should also have enough power left to keep the clock running for months or even years. There is probably a problem with your internal battery, or a contact to it.
The best option is to have it fixed, or to fix it yourself.
You can see the battery is step 7 on this site:
The battery is a MS518SE 3 Volt Lithium battery.
Most cameras have a capacitor or internal battery that keeps the clock and settings active while the battery is being changed. If the camera is left with a dead battery for a very long time, the capacitor or battery may become fully discharged and dysfunctional.
There is usually an internal battery/capacitor that helps maintain small bits of information like date time and location. Most likely this battery is dead and causing you to re-enter the information. The D5100 is not really user serviceable and I wouldn't recommend changing this yourself. You can get yourself a battery grip and get a longer shooting period without having to re-enter this information on shoot days.