Recall that the stomach's chief acid is hydochloric acid - HCl. Therefore, it stands to reason that losing stomach contents causes low levels of both H+ and Cl- ions in the blood. (Note: indirectly, of course. Blood levels of these ions drop because these ions are taken from the blood to make more stomach acid as a replacement.)
But why the hypokalaemia? You do lose a little potassium with the gastric fluid, but so little that it would take 30 to 80 L of vomitus to achieve the level of hypokalaemia seen in these patients! Clearly there is some other process going on, and as it turns out, the guilty organ is the kidney.
Recall that in the distal convoluted tubule, sodium may be absorbed in exchange for either potassium excretion or hydrogen ion excretion. (Actually, this is a two step process: first sodium is absorbed in exchange for potassium secretion, and then potassium can be reabsorbed in exchange for hydrogen ion secretion.) In cases of low H+, there are less hydrogen ions available to swap with sodium, and so more potassium ions must be excreted in their place. (Note that this cuts both ways - even if the primary insult were low potassium levels, as it is in other conditions, the end result would be the same: hypokalaemic alkalosis).
This may seem like a bizarre thing for the body to do - why not just reabsorb less sodium and thereby avoid this entire debacle? You may have a point, but the body has bigger fish to fry. Firstly, it needs the sodium to be reabsorbed, because significant vomiting often implies hypovolaemia. When ever you are short of water, your kidneys should reabsorb sodium, since it drags water with it by osmosis. Reabsorption of sodium is even more crucial at the distal convoluted tubule site, since the concurrent hypochloraemia makes it hard to reabsorb it proximally (where the two are reabsorbed together). Secondly, it is really important in an alkalosis to be absorbing as many hydrogen ions as possible, even if it means you have to swap them for a few potassium ions.
Hope that all made sense!