Here's the chain of events:
Janeway offers to spring Paris out of a rehabilitation camp in exchange for his expertise in finding the Maquis and his knowledge of the Badlands. She commissions him as an observer, much to Paris's objections as "being the best pilot [Janeway] could ever have".
Once aboard Voyager, Paris has no post on the bridge: Lt. Stadi is given the helm at the start of the Badlands mission.
Voyager is thrown into the Delta Quadrant by the Caretaker array. Stadi is killed, as are dozens of other Voyager crew members. Paris survives.
In the ensuing chaos, Paris moves to help secure the bridge and helm, taking the initiative to stand in for Stadi.
The Caretaker beams the Voyager crew and the Maquis to the array in a simulation, with Paris acting as Janeway's muscle in the simulation and on the bridge during her standoff with Chakotay.
Paris insists on joining Janeway to find Ensign Kim on the return away mission to the simulation.
Aboard Voyager again, Paris is seen at the helm when Neelix is introduced, and for the rest of the episode.
Paris joins Janeway on all subsequent away missions in "Caretaker", and winds up saving Chakotay's life in a hairy spot.
At the end of "Caretaker", Janeway grants him a (field) commission in the following exchange:
Janeway: Mr. Paris, you have a problem. I've invited Chakotay and the other Maquis to become part of this crew. It seemed the only reasonable thing to do under the circumstances.
Paris: Will you provide a bodyguard for me, Captain?
Janeway: It seems you already have one.
Paris: I do?
Janeway: Mr. Chakotay said something about his life belonging to you? He'll be taking responsibility for your safety.
Paris: I think I'm going to enjoy this.
Janeway: Don't be so sure: he'll also be my first officer. Everyone aboard this ship will report to him, including the lieutenant assigned to conn.
Janeway: I've entered into the ship's log on this date that I'm granting a field commission of lieutenant to Thomas Eugene Paris. Congratulations.
Paris: For the first time in my life, I don't know what to say.
Janeway: You've earned this, Tom.
So, the three factors that seemed to have given Paris his position were:
- His established ability as a world-class pilot, possibly the best one on Voyager
- The death of much of the Voyager crew, including the previously established conn officer, Lt. Stadi.
- The earned trust of both Janeway and Chakotay during the Caretaker mission due to his initiative, ability to function under pressure, and loyalty
Given these factors (and the fact that Robert Duncan McNeill had a starring role, of course), Paris was the obvious choice.