How did Klink ever become a colonel?

This is to build on Jake's excellent answer.

The "average" American officer basically tops out at Major, e.g. Major Nelson in "I dream of Jeannie." Around the age of 40, an American Major has two choices:

1. He can leave the army on half pay as Major.
2. He can remain a "career" Major, and be rewarded with a final promotion to Lt. Colonel, around age 50, basically for purposes of calculating his pension, and retire shortly after.

A few of the more able or fortunate ones are promoted to Lt. Colonel between the ages of 34-39, and these are the officers that will eventually be promoted at least to (full) Colonel, and possibly General.

From Jake's post, Colonel Klink was born in 1894, was promoted to Captain at age 26 in 1920, and Major in 1924, at age 30, a year or two ahead of the American schedule, If he was promoted to Lt. Colonel in 1929 at age 35, he'd have gotten one more promotion than the average American officer. His promotion to full Colonel, some 13 years later, smacks of an age 50, "final" promotion. All this is because of the following:

In the 1920s, the German army was limited to 100,000 men (only 4,000 of them officers) by the Treaty of Versailles. These men formed the core of a larger army in the 1930s, when Hitler came to power. Because there were so few of them, those men enjoyed unusually rapid promotion through the officer ranks (or to non-commissioned officer, for enlisted men), when the German army expanded.

In choosing whom to retain, the German army selected the most fanatical and dedicated officers, only some whom were were competent, but who were most likely to stay with the army for "life" instead of bailing out by age 40. Klink fit perfectly the mold of a "loyal lifer." This is in contrast to a certain "bailee," a corporal named Adolf Hitler who left the army, pursued a career in politics, and became Commander-in-Chief by that "roundabout" route.

If he had been any good, Klink would have been in line for another promotion by 1934-35 (when the German army expanded more than fivefold, to 550,000 men, in preparation for the 1936 invasion of the Rhineland). Then he would have been "off to the races" given the start of World War II. Two of his contemporaries were Erwin Rommel and Walter Model, who became Field Marshals, meaning that Klink was in the right place at the right time. The fact the he was repeatedly "passed over" at such a critical time suggests that he was "bottom of the barrel" among his contemporaries.

So, too, is the fact that he was given a home command as a prison camp commandant, rather than a field command, attacking other countries. Commandants of bases or other installations are typically Colonels, or at least Lieutenant Colonels. In the case of an "up and comer," such a "desk job" is a test of their administrative ability, and if they do well, they often become generals if and when they are rotated back to the field. But for Klink there was no "when," and it was his terminal, dead-end position.

For someone of Klink's limited ability, "80 percent of life is showing up." He didn't make it to the top, but he got to a relatively high position as an officer in the German army because there were few other candidates competing with him.

According to the Hogan’s Heroes Wikia site page on Colonel Wilhelm Klink his ascent through the ranks went as follows; emphasis is mine and parentheses English rank equivalents as well:

He became a Hauptmann (Captain) in 1920, Major in on December 25, 1924, and finaly Oberstleutnant (Lieutenant Colonel) in 1929, a rank he would still hold when made Kommandant of Stalag 13, until he was promoted to full Oberst (Colonel) in 1942, after the pilot episode, which I discovered. This was due in part to him having an efficiency rating just a few points above miserable.

Yes, no specifics as to how this rank was earned are provided but the implication is that he barely made it to full Oberst (aka: German equivalent of “Colonel”) rank when the series came out of the pilot episode and officially began.

In the episode "Easy Come Easy Go", which was in Season 6, Episode 15, this question is sort of answered. After Hogan's meeting with the "Brass", in England, the next scene is the next morning in their room. Hogan, and Klink were talking, and that's when Hogan asked Klink "How did you get to be a Colonel?" Klink started answering, " My brother's wife knows....", and that's where he was cut off. This basically implied that Klink was promoted to Colonel due to who he knew, or through his family connections.