As with any surplus rifle, check the bore condition (though I've NEVER heard of a bad K31!), smoothness of the action, the trigger (real Swiss precision!) - all the usual things. The metal finish may be worn, and some fairly major scuffing of the buttstock are standard on these rifles - no worries.
If you get lucky, you may find a paper scrap with the 'prior owner's' name, address and unit printed on it. Some new owners have contacted these old troopers, and learned their rifle's history! Try that with a Mauser or Mosin-Nagant! :-)
From what I have read, walnut stocks were common on pre-war rifles, with the beech stocks coming in later. Mine is a '48 model, so I have the lighter beech.
You'll love this rifle - it's by far my most accurate Surplus Rifle.