Why Teamviewer keeps running in the background?

Because that is the nature of teamviewer: it is build to respawn.

/opt/teamviewer8/tv_bin/script/teamviewerd.sysv is probably responsible for it.

To stop teamviewer use:

sudo teamviewer --daemon stop 

It will show ...

initctl stop teamviewerd
teamviewerd stop/waiting

and it is gone ...

rinzwind@discworld:/opt/teamviewer8/tv_bin/script$ ps -ef|grep teamviewer
rinzwind 12712 12428  0 18:11 pts/0    00:00:00 grep --color=auto teamviewer

Commands to manipulate the daemon:

teamviewer --daemon status        show current status of the TeamViewer daemon
teamviewer --daemon start         start TeamViewer daemon
teamviewer --daemon stop          stop  TeamViewer daemon
teamviewer --daemon restart       stop/start TeamViewer daemon
teamviewer --daemon disable       disable TeamViewer daemon - don't start daemon on system startup
teamviewer --daemon enable        enable TeamViewer daemon - start daemon on system startup (default)

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Regarding comment:

From teamviewer 9 help:

$teamviewer --help

 TeamViewer                      9.0.32150 

 teamviewer                      start TeamViewer user interface (if not running) 

 teamviewer --help               print this help screen 
 teamviewer --version            print version information 
 teamviewer --info               print version, status, id 
 teamviewer --ziplog             create a zip containing all teamviewer logs (useful when contacting support) 

 teamviewer --passwd [PASSWD]    set a password (useful when installing remote (ssh) 

 teamviewer --daemon status      show current status of the TeamViewer daemon 
 teamviewer --daemon start       start      TeamViewer daemon 
 teamviewer --daemon stop        stop       TeamViewer daemon 
 teamviewer --daemon restart     stop/start TeamViewer daemon 
 teamviewer --daemon disable     disable    TeamViewer daemon - don't start daemon on system startup 
 teamviewer --daemon enable      enable     TeamViewer daemon - start daemon on system startup (default) 

Options are still there in TV9.

If you install Teamviewer 10 from the .deb file, --daemon stop as recommended above won't work in Ubuntu 14.04 LTS (and others?). Don't know why.

The daemon is an 'upstart' job, so it gets controlled from /etc/init/teamviewerd.conf not /etc/init.d.

Unfortunately, for me, both initctl stop teamviewerd and service teamviewerd stop result in:

initctl: Unknown instance: 

teamviewerd.sysv gets installed in /opt. So, to stop it, you need to do:

$ sudo /opt/teamviewer/tv_bin/script/teamviewerd.sysv stop

systemctl stop teamviewerd.service
systemctl disable teamviewerd.service