In Alfred's very first Golden-Age-cum-Earth-Two appearance, this wasn't the case; Alfred was portrayed as having contacted Bruce asking for a job when the latter was already an adult (and already Batman), with no indication their families ever knew each other. Nowadays this version is totally unthinkable.
The first bit's right, but the second isn't: in his first appearance◊, Alfred says "You may remember my father, Jarvis, who was your father's butler for many years!" (Batman #16). This is why Bruce feels obliged to accept his offer, even though the last thing he wants is a servant poking around Wayne Manor and finding the secret tunnels. The whole "son of Thomas Wayne's butler who became an actor, but felt compelled to join the Waynes after his father's death" storyline is already in place in that comic; just when it happened is different.
Pulled this from the Alfred Pennyworth entry:
The first bit's right, but the second isn't: in his first appearance
◊, Alfred says "You may remember my father, Jarvis, who was your father's butler for many years!" (Batman #16). This is why Bruce feels obliged to accept his offer, even though the last thing he wants is a servant poking around Wayne Manor and finding the secret tunnels. The whole "son of Thomas Wayne's butler who became an actor, but felt compelled to join the Waynes after his father's death" storyline is already in place in that comic; just when it happened is different.
Edited by DaibhidC