You shouldn't need to worry about threading for a simple chat application. The reason I say this is because chat posts are usually pretty small and once they are on the server anyway, the process of writing to a database will result in a transaction lasting milliseconds. This wouldn't affect user performance at all, so the effort may not be worth the benefit.
However, if you want to be super efficient, then threading is a way to go about this. You could always send the data asynchronously to the database once the server receives it in which case the user will see no impact to performance at all and the server will just hum along happily.
You can read this
] to learn the basics of threading in C#