Binary Clock - battery powered and rechargable?

I am new and inexperienced with regards to Arduino and have started with a basic binary clock. I'm following instructions for one which is plugged in to the wall for power, however I would like mine to be free standing. I was wondering how easy it would be to power the arduino with batteries and then when it is plugged in, this would recharge the batteries (like a phone). How would I do it? I thought it might be easier and more professional looking than having to fit some sort of battery compartment. I also though maybe I could incorporate some sort of blinking red light when it was running low on charge?
Thank you!

The plug & recharge part is easy, compared to building a system that will run a reasonably long time before the batteries need recharging. The standard Arduino, typical programs and most displays (especially ones that emit light) are relatively power hungry, so batteries don't last long. It takes a fair amount of technical expertise to get around those limitations, so this is not a beginner's project, but you can read more about power saving techniques with the Arduino and decide for yourself at Gammon Forum : Electronics : Microprocessors : Power saving techniques for microprocessors

Ah ok thanks very much!!