I've brined, injected, and brined and injected chickens. In the past I've only brined turkeys. Doing both injection and brining is great for a short brine like 4-8 hours. I just inject the brine then let it sit in the brine for the period desired. It's a bit time consuming if you have a cheap, single point injector like I do.
For longer brines it really isn't necessary to do both unless you want to guarantee your brine's flavor profile permeates all the meat. I just finished smoking an unbrined, 100% natural turkey we intend to use for soups and you can really tell the difference between a brined one and a non-brined one, especially if your brine has an enhancing flavor profile. I use brines that have a fruit juice base, salt, sweetener, spices, and a little curing salt, but just a little. I like the slightly hammy flavor it gives the birds, a flavor that goes well with the smoke. I plan on doing a 36 hour brine starting Tuesday night so I won't inject. Or, I just might use the injector needle to merely poke holes in the breast without injecting. Hmmm, just thought of that. Have to think about it.