Reinstall OS X 10.10.4 onto external thunderbolt SSD

Yes, that's simple to do. You will need the Yosemite installer application downloaded from the App Store and sitting in your Applications folder.

Partition and Format the hard drive:

1. Open Disk Utility in your Utilities folder.
2. After Disk Utility loads select your SSD (this is the out-dented entry with the mfgr.'s ID and size) from the left side list. Click on the Partition tab in the DU main window.
3. Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Partition button and wait until the process has completed. Quit Disk Utility and return to the main menu.

Install OS X:

1. Open the Yosemite installer application.
2. Click on the link, "Show more disks", below the default disk that is shown.
3. Select the icon for the SSD.
4. Click on the Install button.

Open the Startup Disk preference panel, select the SSD, then click on the Restart button.

Unless you absolutely need the hard drive space, I would leave the OS installed on the internal drive, so that you have an alternate means of starting the computer if the need arises.

You should be able to move files between the 2 drives.

Once you get this set up, do a backup, preferable 2 separate ones on 2 drives.

You cannot easily remove OS X from the internal drive. You would have to erase the drive to do that completely. It may be easier to simply transfer what you want from your Home folder to the SSD then erase them from the internal drive. Leave OS X there in case you should require it in an emergency or it you need to do maintenance and must boot from another device.