How to create a system tray application that appears and alerts to a group of user when a new record is added to database?

There is a SqlDependency class[^] which provides functionality to get information about data add/change events.

Please, see:
Using SqlDependency for data change events[^]
Detecting Changes with SqlDependency[^]
SqlDependency in an ASP.NET Application[^]
Using SqlDependency in a Windows Application [^]

To be able to create "system tray application", i'd suggest to use NotifyIcon[^]

You already have divided your problem into smaller parts.
Now find a solution for each part and then put everything together.

1. How to get User Data from the Active Directory[^]

2. Create a Stored procedure or SQL query to check the table that has changed.
For example you can check if row count of the table has increased compared to previous query.
C# SqlConnection Example[^]
SQL Example:

SELECT COUNT(*) FROM TableName;

(There are plenty of other ways to do this, just do some research.)

3. Not sure what you want to do here.
How do you define the group?
This can be probably be done by a client that each user runs, that polls the database for updates separately.
Or if you have a server you could create a WCF service that pushes information to connected clients.

4. For creating an application so it appears with a tray icon, you can read about it here
C# Application System Tray Icon[^]