I think you're making this harder than it is ;-}
First you'll need a means to connect your new drive to your system. If you can locate a FireWire case to hold your new drive it will be the faster way or you'll need is this: SATA to USB 2.0/3.0 Converter for SSD / HDD.
Before you start you should make sure your firmware is up to date follow this Apple T/N: About EFI and SMC firmware updates for Intel-based Mac computers. If you can backup your internal drive to an external drive this would be a good time to do it (if you think the drive is about to fail). If you can using a different system see if you can prep the drive up as I outline below so you can use it to do diagnostics of your current internal drive.
Download from the Apple Apps store the OS installer onto your still working HD El Capitan. I do recommend making a copy of the OS installer as during the installation process it will be erased.
Now connect your new drive to your system and open up Disk Utility you'll need to create a GUID partition map and create a Mac OS Extended (Journaled) partition. Once done you can then launch the OS installer and select your new drive to install the OS. Create a new user account (different from what you currently have. At the end of the process you will be asked if you have anything that needs to be migrated from another system in this case select your internal HD and the installer will launch the Migration tool and copy over your current user accounts, apps & data over to your new drive. It will also mark off what apps need to be updated.
Once done you should be running from your new drive, if you are using the USB adapter it will be slow! So don't panic! Once everything is OK shut down your system and swap out your HD's and you're done!