The rulebook states that when a character enters the Crown of Command, any characters killed result in the characters' players being out of the game.

But the Dragon expansion and some alternate endings can result in characters on the Crown being killed.

If that is the case, what happens if only one player is left?

The rulebook applies to the standard ending for Talisman, that being the Crown of Command. Other endings from expansions have different rules and conditions.

True, but this is from the Dragon expansion's rulebook:

"Once any character has reached the Crown of Command and confronted the Dragon King, any character who is killed automatically loses the game. Note that this rule stays in effect for the rest of the game."

Therefore, if only one character remains, would he have to continue playing the game on his own despite some cards and spells that depend on other characters?

True, but this is from the Dragon expansion's rulebook:

"Once any character has reached the Crown of Command and confronted the Dragon King, any character who is killed automatically loses the game. Note that this rule stays in effect for the rest of the game."

Therefore, if only one character remains, would he have to continue playing the game on his own despite some cards and spells that depend on other characters?

If he finds proper to play Talisman alone, he can go on and see if he can win against Dragons.

Otherwise the players may agree that he would eventually win the game (= he's declared the winner) or that the game ends in a draw (= the board wins).