Use and charge a Lithium battery at the same time
What happens is pretty much governed by Kirchhoff's current law.  Consider the point at the switch, where the wires from the charger, the battery and the load all meet.
Kirchhoff's law states that the total current going into that point must always equal the total current going out.  It's common sense, really - if the law weren't true then electrons would be piling up somewhere.
So if the charger is producing more power than the load needs, then the excess goes into the battery.  If not, then the battery will power the load.
If you're not using a proper charge controller, make sure you use a &quot;protected&quot; LiIon battery.  Or get a good fire extinguisher.