Always have a goal in mind when digging in the snow. Otherwise you are wasting your time and likely going to get biased results.
Targeted observations include:
Layers of Concern: Generally identified by a local forecast. This is also part of your tracking of the seasonal snowpack history.
Persistent problems: Are the grains rounding (getting better)? Where are they in relation to the overriding slab?
New/ Old interfaces: How well are the two layers bonding? Test slab development, shear strength, compressive strength, and the ability for the new snow to propagate across the slope.
Spatial variability: How does a layer of concern live amongst the terrain? Note areas with more deposition due to wind and gravity. Note areas where the slab is likely to be thinner, or the surface is stripped from wind. Always assume that the snowpack does not lie evenly across terrain, and investigate the ground-truthed differences.
Height of snow: Often underestimated with relation to the goal of digging. Most folks want to get their blade in the snow quickly. But just by feeling for strong over weak snow, and height of snow differences you can denote a lot about spatial variability.
Your goal should be a mix of your own opinion and an expert voice. Avalanche forecasts typically offer advice on what to look for when traveling in the backcountry on a given day. Consider this as part of your trip plan. Your goal of digging in the snow should include verification of the day's advisory as well as data to trend for seasonal snowpack history.
Safe: Never dig in or under avalanche terrain. Safe site selection is paramount. Look for test slopes that mimic the terrain you are trying to travel through.
Relevant: Make sure the area is an accurate representation of the slope you wish to examine. Keep the aspect and elevation the same. Do not worry as much about slope angle as most snow tests are designed to work on flat ground.
Undisturbed: Don't dig in areas with ski tracks, tree bombs, driplines, or areas of previous snowpit profiles. Though, if you are digging in glades, know that your best representation will likely break a few of the above rules which is okay.
Considerate: Do not dig in the middle of common runs, or on the skin track. Always backfill your holes so people do not accidentally fall into them.
Once you have a goal in mind it is time to implement it in practice.
Start by locating the area that will be representative to the slope you wish to examine. Have a probe accessible and start to get an idea of the average height of snow (HS).
Once you have determined that you have an appropriate site, probe the area to make sure there is no intruding ground cover that will skew test results.
Then, mark your area (this can be just enough to fit yourself into and run a few tests) and begin to excavate as you would rescuing a buried victim.
Take a step or two back and dig in towards the hole. Don't go too big or you will be wasting time and energy.
Clean your pit wall with a flat shovel blade and take a step back to observe on a macro level.
Step in and start to feel the snow, observing differences in layers and your layer(s) of concern
Run through a hand hardness test and look for strong over weak layers
Run targeted snowpack tests on the layers of concern
Record your observations to share with the local forecast center, and to trend for the season
Backfill your pit!