I think there are two sides to cleaning a French press: the metal filter which may have some grounds stuck onto it and the bottom of the brewing vessel.
To clean the metal filter, I unscrew the plunger and wash the metal parts that hold the filter with a bit of soap and water (or put them in the dishwasher). The metal filter is more difficult because there might be grounds that don't wash off easily. To get most of the grounds out, I apply a bit of soap and then hold it under a steady stream of running water. I think the soap holds onto any coffee residue and then the force of the water water pushes it all out.
To clean the brewing vessel with the coffee grounds, I add more water so the grounds get mixed with the water. Then I dispose of everything in a mesh kitchen strainer. If some grounds remain in the beaker, I hold it under the running tap, the water stream dislodges the grounds and it's all disposed in the strainer. When the grounds are all in the strainer, there may still be some oily residue in the beaker, just clean that off with some soap and water (or use the dishwasher).
Of course, now you've just moved the grounds problem to the strainer. Luckily, the grounds dry easily because the water drips out underneath. When it's relatively* dry, you can easily remove most of the grounds into the garbage / compost bin. The few remaining grounds which may be stuck to the strainer can be removed similarly to the way I described cleaning the metal filter (just rinse under the tap). This should be a bit easier because you can use a relatively coarse strainer for this.
* If you don't want to wait, rapid vertical movements (open side facing up) with the strainer remove enough water for the grounds to come out in one piece.