The iPad will remain on when it is charging. If you power down and recharge, most times the iPad will start up on its own, so it is not uncommon for the iPad to be powered up when charging.
What you are seeing with regard to the battery gauge isn't right though. Have you done a charge cycle to calibrate the battery gauge?
Use Your iPad Regularly
For proper reporting of the battery’s state of charge, be sure to go through at least one charge cycle per month (charging the battery to 100% and then completely running it down).
A charge cycle is when you charge the battery all the way, and then let the battery run down and the iPad shuts off by itself, then recharge it all the way to 100 % again.
Use the iPad like you normally do and at the end of the day, let it recharge overnight. In your case it may only recharge to 80% or whatever reading you are currently getting, but it should recharge all the way to 100%. Use it until the iPad battery runs down and it shuts off by itself. Then plug it in again and see if it will recharge to 100% like it should.
Both of these are worthwhile reading.
The quickest way (and really the only way) to charge your iPad is with the included 10W or 12W USB Power Adapter. iPad will also charge, although more slowly, when attached to a computer with a high-power USB port (many recent Mac computers) or with an iPhone Power Adapter (5W). When attached to a computer via a standard USB port (most PCs or older Mac computers) iPad will charge very slowly (but iPad indicates not charging). Make sure your computer is on while charging iPad via USB. If iPad is connected to a computer that’s turned off or is in sleep or standby mode, the iPad battery will continue to drain.
Apple recommends that once a month you let the iPad fully discharge & then recharge to 100%.
How to Calibrate Your Mac, iPhone, or iPad Battery
At this link http://www.tomshardware.com/reviews/galaxy-tab-android-tablet,3014-11.html , tests show that the iPad 2 battery (25 watt-hours) will charge to 90% in 3 hours 1 minute. It will charge to 100% in 4 hours 2 minutes. The new iPad has a larger capacity battery (42 watt-hours), so using the 10W charger will obviously take longer. If you are using your iPad while charging, it will take even longer. It's best to turn your new iPad OFF and charge over night. Also look at The iPad's charging challenge explained http://www.macworld.com/article/1150356/ipadcharging.html
Also, if you have a 3rd generation iPad, look at
Apple: iPad Battery Nothing to Get Charged Up About
Apple Explains New iPad's Continued Charging Beyond 100% Battery Level
New iPad Takes Much Longer to Charge Than iPad 2
Apple Batteries - iPad http://www.apple.com/batteries/ipad.html
Extend iPad Battery Life (Look at pjl123 comment)
New iPad Slow to Recharge, Barely Charges During Use
Tips About Charging for New iPad 3
How to Save and Prolong the battery life of your new ipad
Prolong battery lifespan for iPad / iPad 2 / iPad 3: charging tips
iPhone, iPod, Using the iPad Charger
Install and use Battery Doctor HD
In rare instances when using the Camera Connection Kit, you may notice that iPad does not charge after using the Camera Connection Kit. Disconnecting and reconnecting the iPad from the charger will resolve this issue.