The Northwest Weather and Avalanche Center (NWAC) issues avalanche forecasts and weather statements for the region on their website here. Remember, however, that NWAC forecasts are issued for terrain below 7000 feet and only from late-November to mid-April. No avalanche forecast exists for the upper mountain. You must be prepared to evaluate the hazards yourself. Look at weather patterns and data from the Camp Muir and Paradise telemetry sites. Extrapolate and make predictions as to what the snow pack might be like up high on the mountain from this data. Reevaluate and ground-truth your predictions as you climb. Remember that you are responsible for your own safety.
Avalanche hazard can change significantly hour to hour in extreme situations. Be able to make an informed decision as to whether certain slopes and aspects are safe to climb on. Climbers who cannot make an informed decision about avalanche hazards should consider getting some training before climbing or going with a guide service.
Anyone traveling in the backcountry in avalanche terrain should carry avalanche safety equipment (transceivers/beacons, avalanche probe, shovel) and know how to use it. Every climbing party should also have the basic skills to evaluate the stability of the snow pack, and know when to turn around. Winter storms are possible at any time of year on Mount Rainier. Be increasingly cautious while traveling below and in steep terrain during and immediately following major storms or significant increases in temperature.
Here are some things to be asking: what's the terrain like - is it steep enough to slide (generally between 30-45 degrees), and if it did slide where would it go (a crevasse, depression, cliff, etc.)? Do you see any signs of instability (recent slides, shooting cracks, whumphing sounds, etc.)? Are there weak layers in the snowpack? Is there a potential trigger (e.g. you, rocks, other climbers/skiers)? Is there a cohesive slab? How deep are your boots/skis penetrating into the snow? Get all the information you can from anyone possible, particularly climbers who have been on the route recently.
Climbers are reminded to consider the avalanche exposure of their camp as well as their climbing route.