How do I get the battery out of my 2003 HD 883?

SOURCE: 883 wont start

The battery might just be done or warn out, if you get lights when you turn the key on, and can honk the horn and everything, then it should click, if you are able to honk the horn and lights etc. and they aren't real weak, when you hit the starter it should tick, if your lights are bright and horn is loud and brake light responds quickly, then I'd ask a Harley dealer about it.

SOURCE: what is the code for the 2003 HD XLH Sportster 883

If this is the original H-D battery in your bike, you've gotten good service out of the battery. Time for a new one. The general rule of thumb on motorcycle battery life is two years although I have seen them last as long as five when kept on a charger designed for this.
The reason for the short life is heat, vibration and overcharging.

Due to it's small size, it is not advisable to charge a motorcyle battery using a standard car charger. If they are charged too quickly, they will overheat. With the standard car charger, even with the charger set on the 2 amp position, the initial charging rate can still exceed 20 amps. You need a charger that is limited to 1.5 amp output. The new "smart chargers" are the way to go these days. The most popular of these smart chargers is the "Battery Tender" but they are also made by Yuasa, and others. They are specifically designed to maintain the battery even after it reaches full charge. A standard car battery charger will overcharge and boil the battery dry once it fully charged. Using the "battery tender", I have gotten five years of service out of batteries in the past.