How would you build a chat system using JavaScript (web based )and cSharp as you backend language?

What do you mean “how”? By doing appropriate amount of work, of course. Do you have any particular concerns we could discuss?

I think the set of tools you want to use is quite adequate to the task. What kind of help you expect to get?

Agree with the previous comment - it's a bit unclear what you're asking us for. You seem to have a team, you seem to have identified some suitable technologies to use to achieve your goal.

So what is the issue exactly? Are you maybe asking how to organise your team or something like that? Are you hoping we'll design the architecture for you? I mean, by all means do some research into modern real-time messaging system design online, there should be plenty of it. But if we actually design your specific system for you then half the point of your project year is gone.

True you haven't built a fully working piece of software before - most students at your stage haven't. The point of these group projects is to see if you can change that.