Play the fool, in a childish or careless manner.
There are several possible derivations of the word 'lark' in this context. 'Larking about' or 'lark about' (sometimes used as 'larking around' or 'lark around') has been used to mean 'getting up to mischief; playing the fool' since at least the middle of the 19th century. At source its origins may well be somewhat earlier than that; how much earlier depends on which of the proposed origins proves to be correct. The principal theories are that either:
'Larking' derives from the Yorkshire dialect word 'lake', meaning 'to amuse oneself'; for example, in a book by William of Palerne, circa 1350:
[He] layked him long while to lesten at mere. [listen to merriment]
'The Yorkshire pronunciation of 'lake' sounds like 'laik', which could be mistaken for 'lark' outside the county.
'Larking' derives from 'skylark' and alludes to the well-known aerial acrobatics of the European Skylark. When they are on the ground these inconspicuous little birds look like what birdwatchers disparagingly call LBJs - 'little brown jobs'. In the air and singing they are transformed into one of nature's wonders - spiralling upward and trilling an exquisite song. Of course, skylarks were the inspiration for the UK's favourite piece of classical music - Ralph Vaughn Williams' The Lark Ascending. They also appear to have inspired sailors to describe lads who played around in the rigging of ships as skylarks. This term appears to have been coined with reference to the earlier name 'mudlarks' - the children who played and scavenged about the shoreline. Skylarks were first defined in a rather unlikely source, The Student's Comprehensive Anglo-Bengali Dictionary, Kanta, 1802:
Skylarking, the act of running about the rigging of a vessel in sport; frolicking.
The use of 'lark' as a verb begins soon after that, as in an entry for 1813 in the diary of Lieutenant Colonel Peter Hawker:
"Having larked all the way down the road."
The first mention that I can find of 'larking about' in print comes from an edition of the American magazine The Living Age, 1844:
One of the young genelmen was called Mr. Larkins, and I'm blessed but the name he hailed by tallied exactly with the cast of his figure-head and the trim of his craft, for he was eternally larking about somut or other, and his very face displayed a mixture of fun and mischief.
Of the two theories presented above, it does seem that the 'skylarking' origin is by far the more likely.
Whatever the origin, 'larking' may well have been the source of the term 'larrikin', that is, 'hooligan; rowdy young fellow' (although there are other theories on that too). Fred Jago's Ancient Language and Dialect of Cornwall, 1882, includes this definition:
Larrikins, mischievious young fellows, larkers.
The term migrated to Australia and New Zealand and was in widespread use there in the 19th century. In the same way that a current UK term for rowdies - 'lads', has led on to female rowdies being called 'ladettes', polite Victorian society down under often complained about the excesses of 'larrikinesses'.
Joseph Wright's Supplement to the English Dialect Dictionary, 1905, cites Warwickshire and Worcestershire sources that defined larrikin as 'a mischievous or frolicsome youth'. While the English West Midlands may not have been the source of the term, the common use of it there no doubt influenced the choice of name of Jethro Larkin, a character in The Archers, the BBC's long-running soap opera (sorry, 'Radio drama'), which is set in that region.
See also: 'as happy as Larry'.