The trials are basically just special things you must do in a match to earn bonus ADAM. For instance x kills with a certain weapon. The easiest one to do, at least when you start, is using your plasmid. You have to hit someone 30 times. (You don't have to kill anyone, just land 30 hits on people.) You have to do this in a public match, as the description says, but its not hard if you can land a single hit on someone. So just use a single plasmid until you get this.