You should be able to do that.
The easiest way to do this is:
1) Transfer (not copy, transfer) your steam game files to the external HDD.
2) Transfer all data you can to your external HDD so that you have no more than about 100GB left on your OS HDD
3) Turn off your rig. Plug in your SSD (keep your current HDD plugged in as well).
4) Boot up your system and use a cloning tool to clone the internal HDD drive to the SSD (most SSD's have this software available on their website, we can help you find one if you need)
*You might need to format your SSD drive before using a cloning tool. If so, go to Windows Device Manager and format your SSD drive in 'nfts'.
5) If the clone was successful, power down your rig and unplug your old HDD...