Are you sure you'll stick with 8? Many that move to Win 8 don't like it and want to go back to Win 7. You may want to make a system image of that Win 7 drive which could be restored to the ssd should you decide you don't like Win 8.
You can't just delete an OS so as Fixer1234 said, you need to format that drive.
There are many ways to wipe the HDD. That is your smallest problem once you have 8 running on the SSD.
Very Important: When you install 8 on the SSD you must disconnect the HDD where 7 resides. If you don't, the 8 bootmgr will end up on the 7 disk and then you are stuck with 7. After you installed 8, you have to change the BIOS boot order to set the SSD before the HDD (which you can reattach after the 8 installation).
Once Win 8 is installed and tested and you are ready to reformat the old HDD, you will do it using Win 8. The easiest way is to use Windows Explorer (My Computer). Right-click on the HDD and there will be an option to format. (I don't use Win 8 but I assume Explorer is still there and formatting is still a context menu item. If Microsoft crapped that up, too, somebody can correct me on this.)
Don't touch the Windows 7 installation. Just create a separate partition for your data. If it's a 1TB drive, just create another, say, 750GB partition for all your data to be used with Windows 8. It will be your secondary drive but you can still boot up to it by choosing boot drive from start-up later on. After creating the partition, take it out first and replace with the SSD to install the Win 8.
If you need to, remove Windows 7 from the boot list, set Windows 8 as the default, and reboot your computer to be sure it boots into Windows 8 directly, then it's safe to format the drive that Windows 7 is on.
If there is other data on the drive create a new partition on the drive in Disk Manager, move all the data to that partition before you format the Windows partiton.
You should be able to do that from Windows 8.
As others have said, you may want to wait a while to make sure that everything in Windows 8 works and that you really prefer it to Windows 7 before you make the final decision.
If you want to try dual booting, you should be able to enable it using EasyBCD, as long as the Windows partition on your hard drive is intact.
For the OP's purposes I would not want a double boot situation. It is too messy to get the bootmgr of 8.1 out of the 7 partitions and to cleanup the bootmgr and the MBR. It can be done, but you really have to know EasyBCD very well.
Unplugging the 7 disk during the 8.1 installation on the SSD is a much easier and cleaner approach.
The only sure way to get rid of something is to get rid of it, meaning you have to get rid of the data = happens through the ancient old way of zero format or unpart your disk > repart it... But, that is a disastrous move, not to be reversed as it destroys all data; Thus, I would recommend that you follow the suggestions given, NOT to do anything drastic, unless certain.
Having two or more OS is not really a problem, generally. If your major OS works fine, what's the ball game?