Oil leaking from right gear shifter?
There shouldn't be any amount of liquid oil in the shifter, just some grease. 

Looks like some grease is getting out of the holes in the rubber grip and onto the outside.

Regular detergent is not strong enough to wash away grease. You need a proper degreaser product. The citrus based ones are quite good, and available at regular hardware stores.

Obviously the rubber grip needs to be replaced. It's letting grease out and - maybe more importantly - water in. With a Walmart level bike, replacing the whole shifter isn't very expensive. Figuring out what exact model the shifter is can be  difficult though.

The rubber grip should not be worn after a year's light use. Perhaps it was damaged by something sharp. You may be able to glue the holes shut with a rubber cement.