Thing about Buk is that his literary forms were so distinct. The novels are all about his experience, and the poems are surprisingly similar to the novels. The short stories are almost invariably about exaggerated and/or outrageous situations that are observed from the outside. You can burn through his novels in a fairly short period of time, so you may want to save a couple (I've read each of them three or four times over the past 20 years) for later. As Elise posted, it really doesn't matter in terms of chronology, as you can clearly tell what part of his life he is writing about.
Maybe try the poem. Again, many of them read like short novel chapters. I highly recommend "Burning in Water, Drowning in Flame" from 1974. It collects three of his early out-of-print works as well as some post-Mockingbird poems. "Dangling in the Tournefortia," "Love is a Dog From Hell" and "War All the Time" are also great. "War" also contains "Horsemeat," a ~20-odd multi-part narrative poem about the track. Great stuff.
To me, the short stories are the weakest (I use this term very relatively) of his material, but don't think that they aren't good, and even great at times. "South of No North" is very good, and contains two older works: "All the Assholes in the World and Mine (the title alone is a masterpiece - a hilarious, absolute must-read)" and "Confessions of a Man Insane Enough to Live With Beasts."
In the end, read them all. Every word he wrote was important.
I understand that this is the last of his books with his alter-ago Hank Chinaski.
I forgot to add that technically, that distinction would go to "Barfly." That was part of the reason that the movie was so difficult to produce; a lawyer claimed that one of his clients owned the Chinaski character (not just the name, but the character as well). This is described in "Hollywood."