The heading says it all, can a PC be a lich? If so, how? The DMG hints that the Book of Vile Darkness contains the secrets of how; there is nothing official otherwise.
Outside materials for particular campaigns, there is only one official way for a PC to become a lich. The Book of Vile Darkness is detailed in the 5th edition Dungeon Master's Guide (p222). It doesn't describe the ritual to any great extent, leaving it largely in the GM's domain to determine what steps a character aspiring for lichhood ought to take. Some guidance can be garnered from the Monster Manual, though, on creating plotlines for your characters to become liches without the book.
Under the normal monster lore, a wizard can become a lich only through a pact with various evil planar entities, as described in the 5th edition Monster Manual:
Wizards that seek lichdom must make bargains with fiends, evil gods or other foul entities. Many turn to Orcus, Demon Prince of Undeath, whose power has created countless liches. However, those that control the power of lichdom always demand fealty and service for their knowledge. (Monster Manual p203, under Lich)
The process to become a lich involves the sacrifice of the soul of a sentient creature:
...the future lich drinks a potion of transformation - a vile concotion of poison mixed with the blood of a sentient creature whose soul is sacrificed to the phylactery. (Monster Manual p203, under Lich)
The phylactery requires regular soul sacrifices to prevent the lich from devolving into a demilich. The heavy measures needed to sustain lichhood are so extreme and repulsive that it takes a truly evil character to embark upon this road...
Blast from the past: Archliches
Assuming you're ok with adapting material for earlier editions, at least in the third and fourth editions there is a good-aligned variant of the lich: the Archlich. No official 5th edition material exists for Archliches as far as I know. The 4th edition source materials for Archliches describe them as choosing lichhood as a means to a noble end, not as an end in itself, and diligently learning the rituals involved as opposed to taking the easy path of acquiring the secrets of lichhood from Orcus or other fiendly creatures. Archliches retain their personalities and memories, and are not mentioned to need soul sacrifices to avoid becoming demiliches.
While Archliches offer a way for non-villainous characters to reach lichhood, lore-wise the ritual to reach this state has to be hard enough to discourage even most talented spellcasters from choosing this route instead of Orcus's support.
According to the user blade, a method for becoming a lich is detailed in the Curse of Strahd campaign materials. I do not know this method personally, because I am going to participate in a CoS campaign myself and don't want to spoil anything about the module to myself. If you have a copy of the CoS materials, you can look it up yourself.
As I understand it, there are two (arguably three) ways to become a lich.
First, as you mention, the DMG's artifact Book of Vile Darkness has a sidebar that explicitly states
The book could hold a ritual that allows a character to become a lich or death knight
So, that's at least a possibility, but not an exclusive one.
Second, in the description of Liches on MM 203, it loosely covers how Wizards can become Liches. This includes
"make bargains with fiends, evil gods, or other foul entities"
"A lich is created by an arcane ritual that traps the wizard's soul within a phylactery"
"A phylactery is traditionally an amulet the shape of a small box ... possessing an interior space into which arcane sigils of naming, binding, immortality, and dark magic are scribed in silver"
"the future lich drinks a potion of transformation -- a vile concoction of poison mixed with the blood of a sentient creature whose soul is sacrificed to the phylactery. The wizard falls dead, then rises as a lich as its soul is drawn into the phylactery"
This definitely produces a lich, but this path is fraught with plot hooks and DM fiat.
Third, and technically not exactly a lich, is a selectable character option. In the Sword Coast Adventurer's Guide, there is a Warlock patron called The Undying.
Heavily lich-themed, you gain things like Undying Nature. This feature means you don't need to breathe, eat, or sleep, and your aging is slowed to 1 year for every 10 you live.