Within Teamwork, you can easily start a conversation with someone on your Teamwork Chat site.
On your Teamwork site, go to the People subsection of the main People area.

Click the name of the relevant user to open their Teamwork profile.

While viewing the user's profile, you will see an option to Start Chat if the user has access to Teamwork Chat.

Once you click the Start Chat button, it will automatically launch the web version of Teamwork Chat in a new tab, opening a conversation thread with the selected user.
If Embedded Chat is enabled, selecting Start Chat will open within an embedded chat window.
In certain supported areas of Teamwork such as task comments, you can also hover over a user's name to access quick profile options. From the profile pop-up, click the Direct conversation option to initiate Teamwork Chat.

For more information, see: Creating a Conversation or Channel