Well, you initially failed to mention the lab coat and the pursuers so the fight breaking out didn't make sense.
Based on what was included I wrote up the following scenario:
WARNING: This Post is LONG!
Ok, since you asked, here’s how I’d handle it. Rather than have a fight break out right away I’d have the following happen:
I’d have all the player characters in the coffee shop doing whatever they do, cleaning up, having a latte and a scone, whatever. I’d then have the bloodied man stumble in. He’s wearing the remains of a business suit, pants and dress shirt only, no jacket, the sleeves rolled up and the shirt bloody and disheveled. Upon entering the shop he falls to the floor unconscious. When he falls to the floor I’d have the vial roll out of his open palm and roll along the floor and end up at the feet of the character most likely to be heroic, or conversely not have it end up with the character most likely to sell out.
After they have the vial and check on the unconscious man I’d let them notice that it appears that the man has been beaten and tortured, his face is bruised and swollen, his wrists and ankles have rope burns, etc.. They’ll find that he has no I.D., no wallet, nothing to let them know who he is or where he’s from. I’d then let them discuss what they think has happened to the man, what the situation is, and what may happen. If they investigate outside there’s no evidence of how he got to the coffee shop.
Here’s what happened: The man can be anything, a journalist, a mid-level manager or employee for the Umbrella Corporation turned whistleblower, an undercover federal agent. It doesn’t matter but you have to decide (You’ll see why later.). What does matter is that he has learned what the Umbrella Corporation is up to and wanted to expose them. Unfortunately, they found out about him. The brought him down to one of those dark little rooms in the basement and tried to find out who else knew about the new virus, but he didn’t tell. With the last of his energy he escapes them and manages to reacquire the vial that he original tried to sneak out as proof. The flight from Umbrella has used up the last of his will and energy, when he can’t stumble on any further is when he enters the coffee shop. Where was he going? That depends on what he is, most likely a police station or newspaper to let the world/authorities know. He’ll never get the chance though, he’s slipped into a coma and he’ll die overnight in the hospital, should any of the PC’s check.
If the PC’s call the cops/ambulance they should arrive and question the PC’s. Of course, they won’t have much info at this point. Before the police arrive they should come to a consensus about what to do with the vial. Ultimately, it doesn’t matter, whether it ends back up with the Corporation, or in a police evidence vault, or still in the hands of a PC, Umbrella Corp. is going to come look for them. Corporate Investigators are going to assume that the bloodied man spilled his guts. Therefore all the PC’s are going to be targets of Umbrella.
I’d then have the PC’s go on with their lives, go home, continue working, etc.. I’d then have Umbrella Investigators visit each one of them separately. The tactics they use could vary from PC to PC. Perhaps they pretend to be police detectives for one PC (This could be effective for the criminal (“You’re on parole, Johnny, don’t want to do nothin’ to jeopardize that do you? Now answer the questions!”) and just use muscle against another. Try to figure out which way it’s best to confront each PC. This could result in individual combat or a situation the players can role-play.
During this period is when the first infection breaks out. This could happen in a variety of ways:
If they PC’s gave the vial to the police the vial could be accidentally opened while being entered into evidence, or opened on purpose (The police may have thought it was drugs and that the man was a victim of drug violence and wanted to test it to verify their theory)
If one of the PC’s kept the vial, it could be broken during a struggle with the investigators from Umbrella Corp.. It’s then likely that a PC might be infected and thus would present a time factor for figuring things out.
If it’s returned to the Corporation then some action by Umbrella would cause the first infection.
These first meetings with representatives from the Umbrella Corporation should also be when clues start popping up. Perhaps the PI writes down the license plate of the car, pulls a favor from a contact at the DMV and runs the plates and finds out that it’s registered to Umbrella. Maybe the criminal, during a scuffle with the investigators manages to pick one of their pockets and finds an Umbrella ID, etc..
After the clues start coming together they should get back together at the coffee shop to share their experiences. Of course, this is when all hell breaks loose and the infection starts to spread. This is where the real action begins and the players need to start fighting zombies. They also should tie together their separate incidents and start going after Umbrella Corp.. Also, at some point it could happen that the PC’s are captured by Umbrella and brought to those same dark rooms that the bloodied man escaped from.
Either way, no matter how they get into the building they should locate information that ties what’s happened back to the bloodied man. They should find his belongings, such as, his:
Identification (That way they know who he was and what he was doing.)
His notes about what was going on.
And possibly, someone to contact (You never know when the PC’s might need some NPC support.).
Also, if they’ve been infected they should locate the cure before it’s too late.
This scenario would provide a nice slow build up to some very intense zombie fighting. This will allow the PC’s to build experience and familiarity with the setting, especially if they aren’t familiar with Resident Evil.
It’s up to you to decide what victory is for the group. If I were running the game victory for the players would be just surviving and not succumbing to the virus. As for bringing down Umbrella, I wouldn’t let that happen. That way, if the players choose, you could run another campaign in this setting. To make sure this last part happens there could be a massive cover up (The corporation has bought off the local authorities.), the contact provided by the bloodied man doesn’t survive (This could happen off screen, meaning that Umbrella paid the contact a private visit.), no one believes the party’s “crazy” story (Zombie? Zombies don’t exist! You people are utterly insane!) or any combination of the three, up to and including all three.
Hope this helps and I hope that it wasn’t too much information. Feel free to modify it for your needs
Good luck with your game and let me know how it turns out either way.