Facebook chat is a great way for people to communicate with their friends who are currently online. However, you may want to disable the feature if you’re not available or don’t want to chat.
This tutorial will show How to go offline in Facebook.
In order to go offline in Facebook or make yourself unavailable for both chat and video calling, click on the icon 
at the bottom right corner of the messaging sidebar and uncheck Available to Chat
Facebook Available To Chat
Bonus:
Go online to chat if you’re available
Click on the ‘available‘ link on the box ‘Let friends see you as available‘ at the bottom right corner of the messaging sidebar.
Facebook Chat Available Link
Or click on the icon again and check Available to Chat
Facebook Unavailable To Chat
