Hello
Earle.
First, the Krieghoff is a high quality double rifle, so you are starting from a good baseline.
Second, I would absolutely insist on a 3 day inspection period for the rifle. If the seller is not willing to allow you to inspect and potentially return the rifle for 3 days: walk away.
Third, a lot of things to look for with a double, are identical to what you would look for with a bolt rifle. How many owners? How many rounds fired? Condition of the rifling? Condition of the throat? Condition of the stock (hairline cracks)? Etc. etc. A double that has been trashed by an uncaring owner will show, just as a bolt action will...
Fourth: three things are really critical, and specific, to doubles:
1) What ammo was shot through it?
Without engaging in the endless debate of whether mono-metal slug still damage doubles or not, I will summarize the situation. Earlier mono-metal slugs such as the non-banded A Square Monolithic, Barnes Solid, etc. DID damage a number of doubles of either older or modern manufacture. It is a fact. Period. It is the reason why bronze is not used anymore, and even more malleable copper or gilding metal current slugs are typically "bore riders" (their overall diameter is the diameter of the bore at the top of the lands, not the diameter of the bore at the bottom of the groove) and they have a number of "bands" that engage the rifling, seal the gases, and allow the metal displaced by the rifling to flow into the grooves that separate the bands. Was this rifle shot with earlier mono-metal slugs? What to look for? Doubles damaged by earlier overly hard mono-metal bullets will show a shadow of the rifling on the
external surface of the barrel that can be seen when looking at them tangentially (this happened to me and this is how I learned the lesson and I know that the issue is real), and some of them will show cracked solder along the barrel rib or the front wedge, that will not be visible with the naked eye but will show up with a jeweler 10x loupe. If any of that is even ever so slightly visible: walk away.
2) Are the barrels on face?
Another way to say this would be: has the action (the "bar") been bent by shooting over-pressure loads? In the old days it could have been cordite loads cooked in the African sun. In the modern days, it would be over-pressure reloads. This is very easy to verify. You need to go buy a pack of cigarette paper, and have a few copier paper sheets in hand. The "face" is the vertical part of the action upon which the barrels breech and the cartridges case heads rest when the rifle is closed. The "flats" are the horizontal parts of the action and barrels in contact when the rifle is closed. There should be no gap between the barrels and the action at the face or on the flats. Period. To verify this, cut a small strip of copier paper and place it against the right barrel breech as you close the rifle. The rifle should close, but it should not be able to close fully, the opening top lever should not be able to snap back all the way to the center of the action. If you pull on the paper, it should rip. Do this again with cigarette paper. The rifle should fully close but the cigarette paper should rip when you pull on it. Do this again with the left barrel. Then do this with the right flats. Then do it with the left flats. If contacts points on the right and left of the face and right and left of the flats are tight enough that the cigarette paper is ripped when pulled from a closed rifle, and if the rifle will barely close with copier paper, then you likely have a winner (assuming the under bolts are good - see next section). If copier paper slides away when you pull: walk away. If cigarette paper slides but copier paper rips: you are in an imperfect but likely OK area...
3) Is/are the under bolt(s) worn out?
The "under bolt(s)" is/are the piece of metal that slide(s) into the "bite(s)" at the rear of the "lump(s)." Some rifles have a single massive under bolt and bite, others have dual or even triple under bolts and bites. Whether a rifle has single or multiple under bolt(s) and bite(s) is not all that critical. Single ones are much beefier than multiple ones, and both have proved themselves utterly reliable. What is important is that there is no play, zero, none, between the under bolt(s) and the bite(s), and that the under bolt(s) actually wedge(s) itself (themselves) into the bite(s). To verify this, close the rifle smartly and observe the top lever. If it snaps back a little short of the center of the action, it means that the under bolt(s) wedge(s) tightly inside the bite(s) before the under bolt(s) bottom out in the bite(s). You have a winner (assuming the rifle is on face - see previous section). If the lever snaps back past the center of the action: walk away. If the lever snaps back to the center of the action, this is still fine, IF the under bolt(s) is/are tightly wedged inside the bite(s). You will know this by opening the action. If the opening lever resists initially when you push on it, then easily slides after an initial higher effort, this means that the under bolt(s) was/were tightly wedged inside the bite(s). This is what you want.
Brand new double rifles (like high-quality brand new double barrels shotgun) are often so "tight" that they are very stiff to manipulate. A broken in double will be still very "tight" but will have worked itself "smooth." This is good. But "smooth" is not "loose." Any looseness: walk away.
Fifth: call Krieghoff with the serial number and ask them what load the rifle was regulated for.
Finally, the asking price is good, but not ridiculously so. It seems to be a rifle worth looking into...
I hope this helps