Does the Brinkmann electric have two grates or one? I know the charcoal model does. If it has two, I would put the turkey on the top grate and then sit a disposable pan on the lower grate to catch the drippings which can be used for gravy later. You want to get the smoking/cooking temp to around 350 degrees. Not sure how much temp control you will have "as is", since the brinkmann electric element is on 100% of the time normally.
If you can, use a "natural" turkey and not an "enhanced" one. You can then make your own brine (and search here as there are a lot of posts on brines and how to make them) which will make the turkey more flavorful and moister. If all you can find is an "enhanced" or processed with a "XX" percent salt or brine solution, then it was brined before it was packaged. DO NOT re-brine an enhanced bird or it will be way too salty. The majority of chicken and turkeys you will find in your average grocery store are enhanced, but either way this will be marked on the packaging.
You can rub, oil, or do a lot of things to flavor the turkey. Just search some other posts and pick your method. You have lots of options....
As to smoke, I would think you could just make a foil pouch of wet chips with a few small holes poked in the top and sit it on the element & lava rocks. Start another thread and let us know how it comes out (photos too!).