Yes, this sounds very do-able.
(The info below is just from viewing the panels, and from previous PT3 db export and import experience. These are NOT official PT3 instructions. PT3 support would be your best bet for assistance if you have further questions.)
In order to change the location where your HH's are stored, from Full Tilt Poker (other sites should have similar navigation):
Options > Hand History... > check the Save My Hand Histories (already done) > click the Change Folder... button > then navigate and select the folder/external drive of your choice to store your HH's. (Basically change the C: drive to your external drive.)
For importing the HH's into PT3:
Import tab > Auto Import tab > select the poker site
from the list below > select the corresponding poker site tab on the right > under Directory Configuration, change the Hand History Directory by clicking the ellipsis button [...] > navigate to the same directory location where you are storing your HH's.
For moving/exporting your existing DB:
select Database > Database Management > click the Export button. This will create a folder within the PockerTracker 3 folder (under C:\Program Files\PokerTracker 3) for each poker site's HH's from your PT3 DB.
Create a New databse:
Database > Database Management > click the New button and select a location on your external drive.
Re-import your old database to your new database:
Import tab > Manual Import > click the Import From Directory button > select the folder for the poker site you want to re-import. (Repeat for each poker site where you play.) (In previous versions of PT3 tournament summary info was not exported, though it may be now.)
If you want to install PT3 on the external drive
(it may run slower externally). While running the install, you will probably have to select a different location rather than the default C:\Program Files\PokerTracker 3 location. (If there is not an easy way to select the location, you'll have to go to PT3 support for assistance.)
This last step may not be necessary as you can probably just install PT3 on both computers, but have the DB stored on the external drive.
The downside to all of this would be if you were ever playing on your laptop without being connected to your external drive, you would not be able to run PT3 or store your HH's unless you reconfigured your options back to the normal locations.
I occassionally play on my old PC that is too weak to run much, so I don't have PT3 on there. After a session is over, I just move the HH's over to my main PC in the appropriate poker site folders. Then import the files into PT3. This may be an option for you, if you don't want to bother with altering your existing configuration. I forgo the real-time info and HUD though.