The /atmel directory is on the root partition. This may not be the best idea. It would be better to extract the tarball to a directory in your home directory, where you have write access. The package will probably produce its own directory, so you could for example cd to to where you want this directory located (e.g. cd ~/downloads ) and then run the tar command:
tar xvzf /mnt/ATMEL_linux_drivers.tgz
The tarball should be extracted then to your current working directory. There is also a tar option to set the target directory. Actually I think that the name of the option is: --target-directory, but please check the tar man page to be sure.
As to the other part of the question, if there is a install script, you need to either A) first make the script executable if it isn't the the "chmod +x <script-name>" command and then run the script "./<script-name" or B) run is like "sh <script-name>"
One final note. If the original tar-ball is located in the /tmp partition, and the /tmp is on its own partition, you will need to extract it elsewhere because the /tmp partition will be mounted with the "noexec" option. This is done because the /tmp directory is world writable and programs should not be located there.