Liam is the leader of your group and thus acts as a sort of tutorial. This raid was his plan, therefore you can talk to him about the various aspects of the situation and have him explain how they work.
There are two kinds of actions you can take - Red and Green. Red actions typically consume a "turn" and move all of the individual aspects of the hostage situation accordingly. Green actions can be performed as often as one likes per turn and it does not spend an action.
On that note, the hacker can turn on a camera for free which displays in the upper right hand corner. Use this to focus on the hallway to keep an eye on the approaching police.
The police will only move 2-3 turns after you've shouted at them to stop or fired at them to drive them back. If you do not do either after they begin to move, they will move forward with each additional action until you do either.
Attacking the police will upset the hostages, driving them further to frenzy mode. Concentrating on hacking will calm them down if they are not already at a certain percentage of frenzy. Be prepared to balance this.
The hostages have a rhyme and rhythm to them. When they stick a leg out or lean against the wall, they are growing more relaxed. The more relaxed they get, the sooner they will try to run from you or attack you. Either way, you're losing a hostage. If they begin to hold their knees and rock back and forth, they're about to snap and will do the same. Use chatting and shouting to manage their behavior in between hack boost charges.
The hostage with the brown shirt mentions he is the leader of that team - his mood and any actions taken on him will affect the whole of the group much more than if you perform an action on any other member.
You can shoot a hostage in the leg to prevent them from running and also send nearby members into a higher percentage of frenzy. You have a few free actions to leave them like this before they bleed out and die. As soon as you see their skin get slightly paler, you should tend to their wound. You only have 2 medkits however, so use wisely.
You can trade the hostages for various things, including another medkit, better network stability (less chance of being counter-hacked for a limited time) or to push them all the way back to the end of the hall. This can be very useful, but should only be used if absolutely necessary.
When you have only one hostage left, the police will move much faster per turn and move after less turns.
You can use the break room to remove a hostage from the situation entirely, but they will not count as an active hostage the entire time you leave them in there. So if you have two hostages and send one to the break room, you technically will only have one hostage which will cause the cops to move in much faster. The hostages will also grow calm and attempt to attack you or run if you let them stay too long. Moving more than one to that room may prove disastrous for you, but I've yet to actually try that.
Fuck the police.