First, using the Tor Browser, download the latest TBB for Linux, for your desired architecture (linux32 or linux64) and language, from here. Unless you have saved anything that you want in the old TBB's root folder ("tor-browser_...") just delete the old TBB folder (and the old .tar.gz archive file, if you saved it).
If you did save stuff, temporarily move it to your ~/Documents folder. If there are Tor configuration options, such as bridges, that you want to migrate, make note of them. Also, if you have bookmarks, back them up to ~/Documents/bookmarks-yyyy-mm-dd.json and then restore to the new Tor Browser.
Now install TBB as explained here. Work as your username, and NOT as root. Briefly:
user@host:~$ cd ~/
user@host:~$ tar -xvJf tor-browser-ARCH-3.6.2_LANG.tar.xz
"ARCH" is the architecture that you selected (linux32 or linux64) and "LANG" is the language. Now:
user@host:~$ cd tor-browser_LANG
If you have created a shortcut for the Tor Browser, it should still work.
Also, I note that there's no particular magic about pasting a new version of a folder over an old one. In particular, there is no guarantee that doing so will overwrite all of the old data on disk. If that's a serious risk for you, you should be using Tails on a machine with no hard drive. Alternatively, you could use TBB on a dedicated machine with full-disk encryption (FDE). Even using Whonix or a Tails VM, FDE on the host is essential.