As @vc-74 said in the comments, "faire un effort" would be used in the action/moment while "faire des efforts" would be used to describe efforts in a mid/long term situation.
The more correct phrasing in your example would be
Il essaye vraiment de faire des efforts pour continuer avec sa vie comme avant
The first one would imply that going on with his life (after a break-up, the death of a loved one, etc.) is a one-shot effort, which is almost never the case.
Also, "fais un effort" can be and is more often used in a joking/mocking manner
"- Tu peux m'aider à ouvrir cette bouteille d'eau ?"
"- Oh, fais un effort !"