Is there a way for a user to calibrate the autofocus on a Nikon D5300?

I"m not 100% sure the focus on my D5300 is as sharp as is can be

No way to calibrate it, that I know of. Have you gone through all the autofocus settings starting on page 184 of your manual?

Short answer: no. The AF Fine Tune feature is only found on the bodies one step higher than the D5300 (e.g. D7000, D7100, full frame models).

You need to run a test. Print out a focus chart and see if your focus is OK.
To do a focus test, you need to mount the camera on a sturdy tripod. The focus chart will tell you if your focus is off. If you don't have a printer, take a picture of some printed text and use the view finder (phase focus) first, then take the next shot using live view (contrast focus). Live view will give the best focusing possible. If there's a difference between phase and contrast focus, then you have an issue which only Nikon can adjust.

If you are trying to shoot with your aperture wide open and/or at the limits of your zoom lens, the image will be slightly soft in focus. If you set your aperture around f/8, things should be very sharp.

If you're using an older lens, you might be experiencing focus shift.