This is one of those grammar rules that only grammarians really understand and that just about everybody outside of grammarians uses incorrectly. I always recommend avoiding the issue by writing around it. After all, if you are grammatically correct, it will sound wrong to most readers, but if you sound right to most readers, the grammar police will come knocking.
For the record, what follows is how the grammar police would expect you to correctly use these two words.
If the word who/whom refers to can be replaced with he or she, then use who.
RIGHT: Who scored the touchdown? (He scored the touchdown?)
If the word who/whom refers to can be replaced with him or her, then use whom.
RIGHT: Dawn asked whom they were talking about. (They were talking about him.)
WRONG: Whom scored the touchdown? (Him scored the touchdown?)
WRONG: Dawn asked who they were talking about. (They were talking about he.)
Writing around who/whom involves replacing that pronoun with a phrase that means the same thing. For example:
RIGHT: Which player scored? (replace who/whom with which player)
RIGHT: Dawn asked which guy they were talking about (replace who/whom with which guy)
My name is Rob Bignell. I’m an affordable, professional editor who runs Inventing Reality Editing Service, which meets the manuscript needs of writers both new and published. I also offer a variety of self-publishing services. During the past decade, I’ve helped more than 300 novelists and nonfiction authors obtain their publishing dreams at reasonable prices. I’m also the author of the 7 Minutes a Day… writing guidebooks, four nonfiction hiking guidebook series, and the literary novel Windmill. Several of my short stories in the literary and science fiction genres also have been published.