Learn English – By which, to which, at which, to whom: are these relative pronouns in adjective clauses

Here is an article about relative clauses that addresses this particular construction:

4. Giving additional information

We use relative pronouns to introduce relative clauses, which tell us more about people and things....

We use who, whom, whose, and which (but not that) in relative clauses to tell us more about a person or thing....

as object of a preposition:

He decided to telephone Mrs. Jackson, who he had read about in the newspaper. That’s the programme which we listened to last night.

• We sometimes use whom instead of who:

He decided to telephone Mrs. Jackson, whom he had read about in the newspaper.

• The preposition sometimes comes in front of the relative pronoun whom or which:

He decided to telephone Mrs. Jackson, about whom he had read in the newspaper. That’s the programme to which we listened last night.

The relative pronoun who(m) refers to animate beings, like people or animals. Personalities, in the meaning it has in your example, means the characteristic traits of a person. A person's personality is not an animate being. You can't take it to the beach, for example, or feed it donuts. The personality of a person is an inanimate object (a thing). The relative pronoun which is used to represent inanimate things.

A defining relative clause gives us extra information which helps the listener to understand which things we are talking about:

• The man who you saw yesterday is my best friend.

Here, who you saw yesterday tells us exactly which man we are talking about. A non-defining relative clause, on the other hand, gives us extra information about something or somebody we can already identify:

• Your father, whom I've known for twenty years, is the most honest man I've ever met.

Here, the relative clause whom I've known for twenty years, is just giving the listener extra information. It is not helping the listener understand which of her fathers we are talking about!

The relative item that can be used for both things and people/animate beings. However it can only be used in defining relative clauses, and not non-defining ones:

• Your father, that I've known for twenty years, is the most honest man I've ever met* [wrong]
• The man that you saw yesterday is my best friend. [correct]
• The sandwich that you bought looked very tasty. [correct]

That cannot occur after a preposition that has been moved to the front of the relative clause:

• The man [of that you're talking] [wrong]

• The table [at which you're sitting] is antique. [correct]

The options for your examples then are:

A Jekyll and Hyde is a person who has two pesonalities, one of which is bad and the other good.

and

He invents a drug which/that can separate them. When he takes the drug, he becomes an evil version of himself, who(m) he calls Mr. Hyde.