Check everything you would as with any other rifle. Below is roughly my protocol (not in any particular order).
-Break down the BCG. Normally I'll pull the extractor but leave the ejector; for that I just press on it to see if the tension feels right. Check the carrier key screws for good staking, and make sure it's not loose.
-Do a light cleaning of the chamber and bore. Lube all the critical areas as you would with a broken-in rifle.
-Cycle the charging handle a few times to make sure everything moves freely. Verify that you can manually lock the BCG back.
-Insert empty, partial, and full mags with both the bolt locked back and forward to verify they seat properly. If you can follow the 4 rules of safety, cycle some rounds manually.
-To the extent it's possible, check the torque on any threaded items (Reciever Extension, Muzzle Device, Barrel Nut, etc.)
-Verify the FCG is working properly (aka click and reset), and that the springs are oriented correctly.
I'm probably missing something, but that's the gist of it. Any part on any rifle could turn out to be defective or improperly installed. So, the more you check the better. Regardless of how your inspection goes, do not trust a new rifle as your first line of defense. I like to have at least 500 rounds before considering a new firearm for personal/home defense. You just never know how it will function until you actually run it.
Now, specifically speaking to recent Colt problems, I've seen a few random complaints.
1. Extraction/Ejection/Feeding issues. One case was a bad extractor. Another appeared to be an issue with the extractor as well, but no definitive answer was provided.
2. Oddly enough, I saw a report from someone inspecting some new Colt carbines who claimed the Barrel Nut was under-torqued. While it will probably still function, accuraccy may suffer.
3. General fit and finish. This has not typically been a strong suit for Colt. However, most complaints in this regard relate to cosmetics, and won't likely impact proper function.
Hope this helps. Congrats on the rifle, and havre fun breaking it in!