What's the best solution?
Right click and click "Open link in new tab" would be the easiest solution. Your alternative is to output the profile view of the avatar then manually add the link to it. Example as follows.
Code:
<a href="[cb:url location="profile_view" /]" target="_blank">[cb:userfield field="avatar" reason="profile" /]</a>
I really don't recommend opening extra windows for internal URLs though. Built in browser back functionality and right click open new functionality covers those usages fine.