As such, the only context given to your actions in the game is the prologue sequence, which attempts to mimic the prologue of the original game. It can be assumed that this version of the game still takes place after the events of the first Resident Evil, and that Leon is again a police officer on his first day in the local force.
"Zombies have overtaken the city. It's now up to you, a sworn officer of the law, to stop them. It's one terrifyingly evil encounter after another. Can you survive the horror? If the suspense doesn't kill you... something else will."
The beginning of the gameplay corresponds to the same point where gameplay begins in the original game, with Leon on the zombie-stricken streets of Raccoon City. However, without Claire Redfield making any appearance in the story, it's unknown if she and Leon ever managed to meet, or if she maybe even died in the car crash seen in the FMV opening to the original game.
Independent of Claire, Leon makes his way to the Raccoon City Police Department, where he proceeds to solve the various puzzles embeded within. Discovering neither Sherry Birkin nor Ada Wong (who are to be assumed dead?), Leon is able to navigate his way through the infested police department without assistance, and discover the Sewage Disposal Plant on his own. Skipping many more of the interactions with NPCs and any confrontations against the T-00 (though he does confront the G-Virus afflicted Alligator), Leon eventually stumbles upon the Raccoon City Underground Laboratory.
Eventually, after continuing to progress through the laboratory despite a total lack of context or direction, Leon confronts the mutated William Birkin (in his "G3 form"), who he defeats in combat. Immediately thereafter, the game abruptly ends, showing a screen with Leon standing next to what appears to be a police car. No further explanation or conclusion is given, and the game simply displays "The End" before returning to the main menu.
Given all this, it is possible this version of the Resident Evil 2 story takes place in an alternate universe where Leon arrives a day late to Raccoon City, where all other survivors he would otherwise meet are already dead or evacuated.