Dell Vostro 5568 - What SSD can replace the HDD

If you tried an M.2 NVMe drive, then it would never work. You need M.2 SATA.

Dell's Owner's Manual states:

Hard drive options

This laptop supports M.2 SATA drives.

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In another section of the Owner's Manual, it shows removal of a 2.5" drive. So, you may also be able to use a 2.5" SSD, such as the one that you asked about.

I would try an M.2 SATA drive first, and use that for your boot drive (operating system). If you need to add more storage later, you can either replace the M.2 SATA drive or add a 2.5" SATA drive.

If you currently have just a 2.5" drive as your sole drive for operating system and storage, you should remove it before installing your operating system to the new M.2 SATA drive or copying your existing system to the new drive. After booting from the new drive at least once you may then reinstall the 2.5" SATA drive to use as storage.

Or, you can remove your 2.5" hard drive and install a 2.5" SATA SSD where the hard drive was and use that as the primary drive. It's your choice, you have options.  The point is that your laptop supports SATA only, no NVMe drives.

Follow the disassembly and reassembly steps in Dell's Owner's Manual.
Look on YouTube for videos showing disassembly of your model. But, be cautious with them, as some use steps or methods that are not appropriate. Use the videos to add to your understanding of what to do, but follow the steps in Dell's manual.

Whenever touching components or working inside a computer, wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. It's a cheap and sensible precaution. Rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard.