If by "trickle charge" you mean "leaving the cell connected to a charge that in a form of another provides the cell with some power (e.g. a small constant current)" the answer is: ***ABSOLUTELY NOT!***

*You can't trickle charge a Lithium cell. **NEVER!***

It is extremely dangerous.

You can find all the details on battery charging in this application note from Texas Instruments: [Battery Charging (SNVA557)](http://www.ti.com/lit/an/snva557/snva557.pdf). 

See also this page on Battery University site: [BU-409: Charging Lithium-ion](http://batteryuniversity.com/learn/article/charging_lithium_ion_batteries).

Relevant excerpt (emphasis mine):

>Li-ion cannot absorb overcharge. When fully charged, the charge current must be cut off. ***A continuous trickle charge would cause plating of metallic lithium and compromise safety.*** To minimize stress, keep the lithium-ion battery at the peak cut-off as short as possible.

Concerning what you are trying to accomplish, since a cell is a two terminal device, you cannot physically charge it and use it as a source at the same time. The current either enters or goes out the positive terminal!

Of course you probably mean "can I leave the battery connected to the charger while the battery is under load?". Although conceptually possible, this is very risky for Lithium chemistries, because they are so fussy about how they must be charged. 

To do that you would need a charger circuit that not only monitored the battery parameters (as every LiIon charger chip does) but it also compensated for any load variation. Therefore usually you end up with charger circuits that disconnect the battery from the load during the charge. 

If the load is to be powered during charge (e.g. laptop PC), the power for the load is drawn from the charger, which acts also as power supply for the load.