Just like buying anything used, what you're looking for is the age of the camera, and how much wear'n'tear or damage you're willing to trade for the discounted price, and how trustworthy you think the seller is. Think of the click count (shutter actuations) like mileage on a used car. At a certain point, it may not be worth it any more given how much remaining life/reliability is left in the camera.
A D610, being less than a year old as of this writing, is unlikely to have a big discount used since it's a fairly new model, is liable to be rare on the used market, and won't have depreciated a great deal from the new price. OTOH, chances are also good someone won't have had it long enough to abuse the hell out of it. If you find one deeply discounted, you may want to do a little research on the seller and run some serial numbers. :)
You do want to inspect the unit in hand (if you can) for damage to the body. A used seller who is utterly scrupulous at pointing out even the smallest faults is likely to be more reliable than one who proclaims the camera's in perfect condition while the pictures tell another story. In vintage camera circles, there are jokes about how on eBay, the phrase "mint condition" often means "there was a camera in the box last time I looked."
You also want to know what comes with the camera. Not all used models are going to come in the box, with a kit lens, a battery, charger, disk and manual.
Figure out what kind of damage you're willing to put up with: scuffs, scratches on the LCD, some scarring around the mount, dust in the viewfinder, whathaveyou. Obviously, if there's any damage to the sensor or mirrorbox assembly, or water damage you don't want to deal with that.
For me, personally, as a nervous-nelly, I tend to purchase used gear from known reputable dealers with return policies. B&H, Adorama, and KEH are all known quantities when it comes to buying used gear online in the USA and their grading is, if anything, conservative. It's more expensive, but the peace of mind, for me, is worth it. YMMV.
With refurbished gear, you're looking, again, at the reputation of the seller, your discount, and service warranty. The warranty is typically why the price will be higher than that of a used camera. If the camera is factory-refurbished, you have the added assurance that the manufacturer is the one who checked out/repaired the unit (although a great many refurbished units are simply cameras that cannot be sold as new; not all refurbished units are necessarily repaired; both my refurbed 50D and 5DMkII had <100 click counts when I received them) and is offering the warranty. Canon and Nikon sell refurbished units directly off their websites in the US, and warranties to go with them. I think Nikon's is 90 days, while Canon's is now 1 year (same as a new Canon).