Learn English – Origin of “On your own head be it”

OP is mistaken about the exact meaning of the expression. It's not about acquiring "new" skills, but about how you got started on the skills you already have. Here's a definition from dictionary.com

cut one's teeth on: to do at the beginning of one's education, career, etc., or in one's youth: The hunter boasted of having cut his teeth on tigers.

It's a metaphoric reference to when a baby's teeth first appear. They grow (cut) through the gums - often painfully, which also gives us the figurative usage teething troubles.

Once the baby has "cut its teeth", it's properly equipped for the all-important "real-world" task of chewing solid food (metaphorically, for tackling more complex problems in professional life, etc.).

As that dictionary example suggests, the expression is often used boastfully/facetiously (in reality the hunter probably started with much less challenging prey, such as grubs, mice, and rabbits).

This is an American phrase, first recorded in the May 1922 edition of Harpers Magazine:

"Mr. Roberts knows his onions, all right."

According to World Wide Words, this had nothing to do with any Mr. Onions, but:

The crucial fact is that the expression isn’t British but American, first recorded in the magazine Harper’s Bazaar in March 1922. It was one of a set of such phrases, all with the sense of knowing one’s stuff, or being highly knowledgeable in a particular field, that circulated in the 1920s. Others were to know one’s oats, to know one’s oil, to know one’s apples, to know one’s eggs, and even to know one’s sweet potatoes (which appeared in a cartoon by T A Dorgan in 1928). You may notice certain similarities between the substances mentioned, most being foods and most having names that start with a vowel.

They contain much of the verbal inventiveness and mildly juvenile wordplay that characterises another American linguistic fad of the flapper period, that of describing something excellent of its kind in terms of an area of an animal’s anatomy (elephant’s instep, gnat’s elbows and about a hundred others — see my piece on bee’s knees for more).

As with bee’s knees, one of these multifarious forms eventually triumphed and became a catchphrase that has survived to the present day.

The Phrase Finder agrees:

Other phrases that refer to knowing - 'know the ropes', 'doesn't know shit from Shinola' etc. allude to specific items as the focus of the knowledge. Other 1920s variants of 'know your onions' are 'know your oil/ oats/apples' etc. The only one that caught on and is still in common use is 'know your onions'. So, why onions? Well, as the citation above asks - why not? Explanations that relate the phrase to knowledgeable vegetable gardeners, or even to C. T. or S. G. Onions, are just trying too hard. 1920s America was a breeding ground for wacky phrases (see the bee's knees for some examples) and this is just another of those.

Edit: A tantalising snippet in Google Books shows this may have been used in 1908 in a humorous poem in The Postal Record (Volumes 21-22 - Page 27). It' shown in the summary, and is interesting as the year 1908 is also shown. Care must be taken with snippets, as they're sometimes incorrectly dated, but here it is anyway:

But, never mind; Billy knows his onions, He Is not troubled with corns or bunions. He travels along at a good, fair gait; Unless the roads are bad, he Is never late. O. 8. WHITE. WHkesbarre, January 1, 1908. West Hoboken, N. J. At the regular meeting of Branch 1065, held on January 10, 1908, it was honored by the presence of Brother Kelly, President of our National Association.