In terms of how you actually do it, you have several options.
1) Pay someone to do it. Not recommended since you can do:
2)
Shapeways pendant creator
Export (or create) your logo from illustrator (or whatever) as a high resolution bitmap. You need it to describe a heightmap, with black being tall, light-grey being short and white being empty space. You also need to take into account the minimum height, because you can't let the print be too thin - effectively your lightest colour (other than white) should be calculated to be at least the minimum material thickess.
Then you can just use the creator. This lets you add a fastening loop, and is quick and easy provided you're happy with a flat or mirrored back. If you want to conserve material (to reduce the printing cost) then use:
3)
shapejs This needs a bit of programming, but if all you need is to make a concave image that's easy enough that I could give you code to do it.
Resolution matters in three ways.
1) Input resolution : you'll only get the detail out if you supply it in the image. You'll see if you have enough when you use the shapeways pendant creator. I would go for much more than 300dpi - at least 600 dpi would be better if you can get it. Assuming you have a vector graphic, this should be straightforward.
2) Printing resolution. You might just be able to get away with printing in steel, but the lost wax process (bronze, brass, silver and gold etc) are rather higher. The cost is, however, rather higher. In particular with steel, the across-face resolution is much better than the depth resolution;
3) Material strength. You need the material to be thick enough to survive the printing process. Also, if you get it polished, small details can be rubbed away.
Best way of determining cost : generate a quick version of the piece in the right dimensions, upload the model to shapeways (make sure the dimensions are correct), read out the prices. They let you do this for free, and you're not obligated to print every version, so don't fret - the only cost is your time waiting for the calculations to complete.