Photo library of 750 GB on a 1TB drive

I both a 5T Seagate external drive. But it seems to want to just act as a mirror back up of files on my Mac. Have I bought the wrong piece of kit?

Is backup software installed on your Seagate volume? Then I would format the volume with Disk Utility (format it MacOS extended (Journaled)), before you move your Photos Library to that volume.

Any software installed on the volume will interfere with your Photos Library. The library must not be on a NAS or any volume shared over the network. Use a volume that is directly plugged into your Mac. Look at the link in the post of rkaufman. It describes the requirements for an external volume: Move your Photos library to save space on your Mac - Apple Support

I would recommend using an external drive rather than Cloud storage. it's a pretty simple process that many users have been doing for years. i have a 2TB's of Photos on an external drive and have been doing this for over 10 years with no issues. Apple explains how simple it is to do in https://support.apple.com/en-us/HT201517

Generally, other than run your Photos library what kine of work do you do with your Mac?

Remember, iCloud Photos is not an off computer storage facility. It's a syncing feature which lets you have the same library on multiple devices and any changes to one device will be reflected on the other devices and in iCloud.

With a 9 year old Mac you're pushing the envelope with it. It's about as old as I am, definitely vintage and in one year will be considered obsolete by Apple. Then you'll have difficulty getting parts if it needs repair.

I'd do what rkaufmann87 suggested and put the library on an external HD, preferably SSD. Also I'd consider getting a newer iMac. If budget is a concern Apple sells refurbished Macs as a discount but with new Mac warranties. That way if and when you get a new iMac you can take the EHD with you. Also, in the spirit of fair disclosure, I'm not a fan of iCloud libraries.

If you consider getting a new Mac do not get a 21.5" iMac. Get a 27" iMac as it allows users to upgrade the memory themselves at less than ½ what Apple charges. Get them from Crucial.com or MacSales.com.

Also get the largest SSD drive the budget can afford. 1TB is a good size. I have a 500 GB SSD but have numerous external SSDs for additional storage and booting into older systems.