Climbing burns up the majority of your time when riding in the backcountry. That means if you make a bad call on ascending gear or route selection, or are just a poor climber, you'll have lots of time to suffer through your mistakes.
It's exceedingly crucial to bring the right tools for the job. If you're climbing a powder-laden ridge that's 35 degrees or steeper, you need climbing snowshoes, ones with a locked-down heel and a downward-curving nose for front pointing. Snowshoes with an upturned nose and a free heel won't cut it. A split board won't cut it on the steeps, either. On the other hand, if you're busting trail through miles of flat, deep pow, anything other than a split board amounts to torture. Finally, in all ascending situations, use four-wheel drive. In other words, never climb for an extended time without poles–it's way too much work and thrashes your knees.
When it comes to route selection, look at a map. Practice seeing the topo lines as three-dimensional land surfaces. Establish the path of least resistance and take it, and don't forget to consider avalanche danger. More often than not, ridgelines are the easiest, safest, and most direct routes, assuming you've got climbing snowshoes. All bets are off if you're post-holing.
Most serious Alpine routes require a closer look before the descent. Riders who have the technical climbing and snow-assessment skills will often ascend their chosen descent to avoid surprises on the way down.
No matter what you've got on your feet, there's one climbing trick you should always employ–the rest step. The key element of the rest step involves locking your downhill leg just after placing your other foot up the hill. Before you move your weight onto the new step, allow yourself to be completely supported momentarily on the fully extended rear leg. Concentrate on letting the bones of this leg support your weight. This allows the quad and hamstring muscles to relax briefly, causing a surge of unimpeded blood flow, which flushes out lactic acid and brings in oxygen. If you do this at every step, it only requires a split-second's hesitation. At extreme altitudes, however, you may want to hang out on your back leg for several breaths.
You should also give your chest, arm, and shoulder muscles their fair share of the work. That means the length of your (hopefully adjustable) poles should be shortened enough to provide good leverage. Keep your palms on top of the poles and your elbows high. Your knuckles should point into the slope, not to the side. If your forearms and elbows are above the pole shafts, you're doing it right. Imagine that your pole shaft is the lower leg, your forearm is the femur, and your hand on the pole grip is the knee joint. By keeping the angle between your forearm and the pole shaft significantly greater than 90 degrees, you'll generate a lot more power. Once you figure out this power-transmission thing, it's amazing how much work you'll spare your legs and how fast you can climb.
The key to good traction and safe, efficient, climbing is to keep the entire surface area of your boot sole, crampon, or snowshoe in contact with the slope. In soft snow you'll almost always be front pointing (stepping straight into the slope with your toes). But on steep, firm snow, try turning your foot across the fall-line. With both feet sideways (perpendicular to the slope) you can use a cross-over step, which takes you either straight up the fall-line or carries you up and forward, depending on how far you cross over. This method is known as the French technique. You'll need to alternate facing left and right every few minutes.
If you remain facing straight up hill, you can also use the duck step (toes out, heels in, each foot at about 45 degrees to the fall-line) to keep your entire foot in contact with the snow. This works until it gets too steep. You'll want to rotate through all three of these methods (front-pointing, French technique, and duck step) in order to not overwork any one set of muscles.
Keep in mind that at some point of steepness, firmness, and exposure, an ice axe and crampons become necessary to safeguard against falling. If learning crampon and ice-axe techniques appeals to you, check out Mountaineering: Freedom Of The Hills by or any other basic mountaineering text at your local climbing shop.