To checkout using ssh and key authentication with tortoise start by installing both tortoisesvn and putty (using windows installer)
Get Putty here
Get Tortoisesvn here
Click Generate and wave the mouse around in the box
Fill in the Red boxes, copy and paste the text in the blue box and then save your private key somewhere safe.
Now take the string you have copied and pasted and insert it as a line in the text file on server
you may then have to change the permissions on this file using
chmod 700 ~/.ssh/authorized_keys
Now run Putty from the start menu like in the first image
and configure it as shown in the next few images.
Click the auth item under ssh and browse to your private key location
Set the auto login username
Set your hostname, name your session and click save.
Now click open and you will be authenticated with your key plus your pass phrase.
If you want to have your pass phrase remembered each time you can now run Pageant from the start menu and add your private key and pass phrase.
This will run in the background and automatically sign you in when putty needs authentication details.
You should now be able to test it by running putty and double clicking on the saved session.
You should now be logged in to the remote terminal.
You can now checkout using the a checkout url similar to this
When checking out you are now challenged for your key pass phrase.
Note: For those who wish to have this passphrase remembered for that login session simply grab your private key file and drop it into your start menu startup folder and on boot you will be prompted for your passphrase and you will remain logged in.