There seems to be quite a bit of confusion concerning the charging of the LinkIt One Li-Battery, and I'll admit it is pretty non-intuitive since for years I knew you should never connect two power sources at the same time to an Arduino. But that is exactly what you have to do to charge your LinkIt One battery. Charging the Li-ion Battery that comes with your LinkIt One is actually a simple process. If you follow these instructions you should have no problems.
Step 1: Set Your Board for Charging
In the image you see the switch marked with the red arrow. That is the switch that allows the board to be powered by either the USB cord to your PC or the Li-ion Battery. To charge the battery slide the Bat/USB switch to the Bat position (that position is the side marked by the red dot) and then connect your battery. Once that's done now connect your board to your PC with the USB cable leaving the switch in the BAT position. And that's all you do. Congrats you're charging your battery.
P.S. The Green dot marks the USB position of the switch. Put the switch in that position when using the USB cable to upload your sketch.