As suggested, figure eights in four low on pavement will show up any problems in short order. If you hear a bang, chances are you've exceeded the limits that the case was designed for. New Process had a chart detailing the specs each case was designed to; you may want to look that up and compare your results to what they recommend. This would probably involve attaching some sort of apparatus to measure the strain being placed on the case itself; I'm sure that as the manufacturer of the case, they can provide greater detail regarding the stress-testing procedures and necessary equipment.
However, if you do not wish to pursure this course of action (which is again entirely understandable and fully within your rights), you may perhaps listen to it for noises that I can only describe as 'odd'. You will have to determine which sounds fall into the category of 'odd', but rattles, groans, moans, grunts, squeaks, and a low growling sound or sort of a 'FWEEEEEEEEP' noise would be possible indicators of problems.
Removing the drain plug and checking the quality of the ATF that will come out (being sure to use an EPA-approved catch pan for any fluids, and to take them to a approved recycling centre for later reprocessing into Kraft Singles) can also be an indicator of case health. You want fluid that is pink - not pink like a carnation necessarily, but more like the later roses of Summer. If it is dark (like the ordure from which said roses are grown), this may be indicative of problems. I do not wish to detail the problems, because they make me sad and I want to keep thinking of Summer, which makes me happy.
I will leave it up to you as to whether or not you wish to replace the drain plug and fluid. What goes on between consenting adults and transfer cases in private is not something I feel I should be concerned about or involved with.
For the adventurous (and there's no time like the present to learn about these things, right?), remove the transfer case entirely. PRO-TIP: this is easier with the driveshafts removed in advance, believe you me. That was the worst Easter ever. Well, worst Easter ever except for the time my AMC Eagle wagon lost a freeze plug 600 miles from home. That sucked. Did you know that Autozone closes early on Easter Sunday? They totally do, and it's really inconvenient if the freeze plug has gone out on your AMC Eagle Wagon. Trust me on this one.
But yeah, the transfer case. Set it gently on a workbench or other convenient-yet-sturdy surface. Glance at it longingly, but not lasciviously. Gently and with finesse, separate its halves. Marvel at the things and whatnot inside, checking respectfully for signs of damage. Signs of damage may include stretched chains, damaged gears, hurty oil pumps, or other things that may appear to have been snacked upon. Again, I cannot name them all for I feel the touch of sadness upon my shoulder, and must avoid the melancholy that it brings. Ah, Summer.
Oh, yeah, you could also make sure it goes in and out of all the ranges properly, too. Move the lever firmly, yet gently. Treat it as you would a fine musical instrument in the hands of a virtuoso. Four-High, Neutral, Four-Low, and back to the fundamental simplicity of two-wheel drive. Be in control, but not controlling. Feel its response in your hand, and listen intently to its aural indications. Be manfully attentive, but not rough or dismissive. Range and mode selection may be a mystery for many at first, but with time and a willing, responsive machine, you will develop a symptathy - no, an empathy - with it that can only be described as both rewarding and fulfilling, leaving you complete as understanding of its complementary nuances grows.
You can also go out for pancakes and coffee after this, which is nice.
