I've struggled to take nice, clear photos of coins. Silver coins seem especially dificult. They can quickly become mirrors. I'm now using a Nikon Coolpix. For close-ups I have rigged up a device to use my OLD binocular microscope. It invloves the macro setting on my camera and a toilet paper tube. Don't laugh it works pretty well. Here's a photo of my set-up less camera and an example of what it can do on a Morgan Dollar. The black spot is an artifact of the microscope. Like I said, it's old.

I started  years ago with a point and shoot.  I then progressed to a DSLR, then a SLT ( sonys fixed miiror technology.  I moved on from the SLR type camera. I am now using a mirrorless interchangeable lens camera. It combines all the advantages of a point and shot with those of a DSLR.  With the absence of a mirror I can use old manual focus macro lenses ( of nearly any make).   My suggestion to any one looking for a  new camera, if you are looking at a DSLR stop. Go and check out what mirrorless cameras are in your budget, then decide. If you can hire a camera to try out before buying do so.   

In regards what do you want to do with the picture after it was taken, do you want to present it in factual research, your personal edification, or a club presentation. Pull the picture up on your computer and find the levels adjustment and play with the settings to fmiliarize yourself with the capability and then go from there.  Go from there, have fun. 

Nice camera. I've never thought of that!

Ian, when you get a chance you ought to show them your bellows set up for extreme magnification. Not just plus lens attachments for the front of the lens. 

The definitive source book is "Numismatic Photography" by Mark Goodman.
He addresses your specific issue.
This book is awesome and (IMO) indispensable for the beginner and the experienced photographer as well.

I do use DSLR's (Nikon D800 and D600) with pro lenses and lighting.  As a semi-pro photographer, I figured "how hard can it be"? 
Well, coin photography is very tricky and Goodman's book steers me right.

Also, have you tried a good quality dedicated scanner?
I stick with Epson for printing and scanning for my clients, as well as personal use.
Scanning at 600dpi furnishes a very accurate surface profile of a coin, but does not transmit luster.  Scanning is important for positive identification (I'm thinking slabs here), especially if coins are lost or stolen.

BTW, I agree with Ian:  Don't buy a DSLR; go mirrorless.
I personally would not replace an already owned DSLR with mirrorless.  In absolute terms, mirrorless technology is still 3 to 5 years away from burying DSLR's, but DSLR's are indeed dinosaurs.

I would have to re enforce what Greg has said about mirrorless and DSLR's. Keep your DSLR for now.  The mirrorless scene is moving very quickly. Sony is bringing out a new version of its A7r in the next few weeks.   It  will have an electronic front curtain shutter( reduces internal vibration) 5 axis in body stabilisation ( X, Y, pitch, yaw,  and roll) a 36mp sensor (the Nikon  D810 uses the same sensor) It will be a phenomenal little camera, but In another couple of years  manufacturers will have taken mirrorless to even higher levels.   So waiting may well pay off

I generally prefer axial lighting, although I do experiment with the location of the light source and switch to copy stand lighting for some pieces. I use Photoshop for post processing. With a little time and effort, you can sandwich different exposures and light angles to create an image that cannot be shot with a single exposure.

Two examples of Canadian coins are attached. The first has two colors of gold overlay. Three images were shot to expose the silver, the gold, and the rose gold. I put them together in Photoshop to show the coin in its "best" light.

The second image is an Olympic "holographic" coin. Colors change as the coin is rotated in the light. I used multiple exposures, each with a different light angle to bring out a different color, then assembled them in Photoshop to mimic the effect of turning the coin in the light.

Digital photography allows one to experiment to create the desired effect if you are willing to take the time.