Sounds to me like it is as simple as replacing your battery! When a battery becomes weak it tends to loose its ability to hold a charge. Definitely not the fuel pump (thank goodness cause there not cheap).
Sometimes just taking a look under the hood can help you determine your situation. Look around the battery for any signs of corrosion. You may just have such a bad connection that your cars alternator can't even think about charging the battery. If this is the case, your battery wouldn't be able to start the car if it wanted to. The terminals should be absolutely clean and so should the cable ends that connect to them.
FreeAutoMechanic.com TIP: I use hot water first to wash off the majority of the corrosion. Just run some water through the coffee pot and dump it right on the battery. This removes enough corrosion that makes it easier to see things and helps cut down on the dust that you don't want to breath.