How would someone get “hired” by a king or a noble? Whether we’re talking someone important like a seneschal, or even just someone like a maid or a cook, how did they come to serve a noble or the royalty so directly?

Well, it depends greatly depending on the region and the time period (was serfdom active at the time, for example). 

But to generalize: kings had lands of their own and would recruit from their population, whether it’s by compelling labor or by paying wages. Moreover, since occupations tended to be hereditary, servants would train up their kids to be servants, so there was an element of self-reproduction.

However, who held the job also depends on the job. Cooks and scullions and porters are clearly peasant jobs, but other jobs like being a seneschal or a chamberlain or a secretary or a treasurer or an equerry would be plum jobs for younger sons of the nobility or the gentry because of the potential for royal patronage. 

Moreover, because of the way in which the person of the monarch was elevated to near-godly status, some of the jobs you wouldn’t expect were reserved for the nobility. So the position of the Groom of the Stool, the man responsible for assisting the king with evacuating his bowels and wiping himself, was one of the most hotly sought-after jobs at court by the nobility, because their literally intimate relationship meant that the Groom would have a lot of face-time with the king.