If I understand you correctly you want to run your OS from the external HD?
this can be done -
Please note that it will be slower if HD is connected over USB, faster over ThunderBolt and always best and fastest from the internal SATA hard drive.
If you want to continue and install OS onto external HD
please restart the Mac and hold CMD+R keys.
This will take you to recovery mode.
Please use Disk Utility to partition and format the new drive.
Run recovery installation by pointing to the external HD.
Once you are done you can boot your computer either to your original internal drive or your new installed external one. You can choose it in Startup Disk in System Preferences.
Post if you have more detailed questions.