Charging and Supplying Power Simultaneously

Most chargers that can supply more current than your circuit requires will charge the battery and supply load current at the same time.

Low cost trickle chargers may not be able to supply enough current.

Some charging arrangements turn off output power while charging to allow trickle chargers to be used.

If you supply more details about your charging circuit and how the load, the charger and the battery are connected, a more helpful reply is possible.

Pulling power from a battery as it is being charged is not really a good idea (especially when you are relying on circuitry that is unknown to you).  A lot (by no means all) battery charger for Li-Ion batteries have a dual mode of charging.  The first phase is constant current, where a fixed current is applied to the battery while monitoring the voltage of the cell.  Once the battery voltage reaches a threshold (4.1v or so), the charger switches to a constant voltage phase, where the charge supplies a variable current that will keep the voltage fixed.  In this mode the current is generally a ramp, moving towards zero, as the cell is completing the charge cycle.  Once the current reaches a minimum level, the cell is considered charged and constant voltage phase is ended.  This constant voltage phase is where the charging while pulling power gets a little dicey.  The charger may not be able to detect when the charge cycle should end, as the loads current draw might be at a higher level than the charge termination level, thus forcing the charge to continue in the constant voltage phase much longer than necessary.  There are documents that suggest that it is in this constant voltage phase that Li-ion batteries a most subject to damage that can eventually limit the charge capacity of batteries, and forcing your batteries to stay in this mode will likely lead to premature failure.

What I try to do in my designs that have embedded rechargeable batteries, is to run the electronics and supply loads from the charging input, not the batteries.  I have blogged on a few of my battery pack designs on this site.  Here are some links:

Smart Solar Lighting Project - Detailed design

Walky the Biped Robot - Power pack

The first link shows a Li-Ion charger, without an output boost circuit, i.e. raw battery voltage output.  The second link is a Li-Ion charger with an integrated 5V boost circuit.  Both of these designs allow for the processor and support circuitry to be powered via the charger input voltage during charging or via the battery when the charger is not connected.

I hope that this will help you with your design.  Please feel free to ask any questions.

You could, potentially, use the power input, if the power supply is sufficient, to directly power the load and also charge the battery.

What I mean is, if you have a phone, when you plug the charger in (I'm not really sure how it works), it starts being powered by the power input while you're using it but also charges battery. Similar to that concept, perhaps.