How to power a portable project with multiple components?
A lithium pack of 4S or 14.8V. Regulate down to 12V and 5V. Switch up the charging voltage as needed. Or take a look at existing devices with the same needs. Any number of iPad information, or Android tablet reference designs. A dedicated PMU, power management unit may be ideal. Look at Texas Instruments website for some ideas.

Also see if the back light can work at a lower voltage, and you may get away with a 3s battery instead. 

Edit:

All iPads use multiple parallel cell 1S Lipo setups. In that case, you regulate the input voltage down and regulate the output up for 5V and 12V. Like any typical usb power bank, reference designs are everywhere.