FAMILY
Starting from the very first games, Bill was one of the few characters who had family members included in canon. With subsequent games, his family expanded to reveal more members, each offering a little more backstory. Strangely, this part of canon never made it to other incarnations, so this is one of those pages that can be considered game-only canon.
Grandfather
The first member of Bill's family revealed in canon, his grandfather first made in an appearance in the Gen I games. He's located in a house in Fuchsia City (next to the Fuchsia Gym) where he'll reveal that Bill had always been a collector, just not necessarily of Pokémon. In Gen II, he took on a more important role. Instead of staying within Fuchsia for the duration of the games, he's now located in the Sea Cottage, where he reveals he's housesitting for Bill. Additionally, he'll ask the player to see certain Pokémon (which he'll then describe). Showing him these Pokémon allows the player to obtain five different evolution stones; this is the easiest way to obtain them in Gen II games (besides recording the phone numbers of certain trainers or trading them from RBY). In Gens III and IV, his roles remain the same, although the Pokémon he asks to see change.
Mother
In GSC, Bill's family extends to add his parents. The first of these characters is his mother, who the player can meet as early as GS. She'll be located in a house in Goldenrod City (just north of one of the entrances to the Underground), where she'll state that her son had gone to Ecruteak for business. Beyond that (and a cute line of dialogue that should give players warm and fuzzies in HGSS), she's more or less a housewife in the games. She spends a lot of her time taking care of Bill's youngest sister (see below) and the house, although according to rumors, she was at one point a kimono girl. What this means for Bill isn't entirely clear. Keeper's chief researcher has placed a bet on "Memoirs of a Geisha totally happened here."
Younger Sister
If you stop by the same house where the mother is located in GSC or HGSS, don't forget to talk to the little girl there as well. She'll offer to give you her older brother's number for your Pokégear. Say yes and surprise! Bill's number gets recorded! Beyond that, Bill's youngest sister is pretty much a typical adorable little kid. She's very perky, and she apparently looks up to her brother very much. (She even brags to you, a complete stranger, that her brother made the storage system.) Of course, one has to wonder about a kid who'd be willing to give away her brother's phone number to people she doesn't even know... or the kinds of calls Bill must get as a result.
Other Sister
Traveling back to Fuchsia in GSC and HGSS, there's a girl who lives in the house of Bill's grandfather. In Gen I (and, by extension, Gen III), she doesn't reveal her relationship to Bill. However, in Gen II (and Gen IV), she states rather plainly that her grandfather is at her brother's house. She doesn't say anything else about him, including whether Bill is her older or younger brother.
Father
The only member of Bill's family who doesn't make an appearance in any of the games. Much more, he's only mentioned in one game: Crystal. In that game, Bill's mother states that his father had been a Pokémaniac and that Bill must have taken after him. At the same time, Bill states that his father has, once again, run off to the Game Corner in lieu of working and that this is apparently a frequent occurrence. (How that family makes money is anyone's guess.) Interestingly enough, this is also the only game to feature a Move Tutor who appears outside of the Game Corner on certain days. Whether or not Game Freak meant to imply that the Move Tutor is Bill's father is not known.
???
In addition to Bill's sister and grandfather, a third character lives in the house in Fuchsia. All this character ever says about Bill is that he catalogues his own Pokémon, so it's not known whether how he and Bill are related, if at all.