For adding profile picture in the facebook chat, all you need do is put the unique id in double square brackets [[ ]]. If you use the name, you might get your friend only if he has a unique name that depends on the facebook database, so it is always advisable to go for the unique Ids.
For inserting my facebook profile picture all you need to do is just type
[[vibudhsingh]] in your chatbox. As my facebook unique id is vibudhsingh.
What is my facebook unique Id?
Facebook allots all its users a unique Id, that is used to identify its user. It can be seen when you go to the timeline or the homepage of that person/page.
Here my facebook id is vibudhsingh which is written just after http://www.facebook.com. Here are few more examples of the facebook unique Id, hope that helps you better to find out your facebook unique Ids.
This same technique is used to insert jumbo sized images in facebook chat: