The answer to this question is subjective and best left up to individual readers. A reader is certainly able to make a case that the character can't be sufficiently modernized for a twenty-first-century audience; although, I think that would be naive. A reason that this play and many other Shakespeare pieces are still read in schools today is because the characters and themes shown are still relevant. The character of Macbeth is a guy whose ambition becomes a more powerful driving force than his moral compass. Adapting that character and that main character trait to a modern audience could be as simple as having Macbeth be the Vice President of the United States. That puts him in a position to be one murder away from being "king" of the United States.

The show Designated Survivor actually explored this exact setup in season one. The assassination attempt wound up being unsuccessful, but the show went so far as to have the Vice President's wife be the driving force behind his actions. There was a scene in which he didn't want to go through with the plan, and she pushed her husband into continuing. The end goal of becoming president was enough to overcome his moral compass. In my opinion, modernizing Macbeth's character basically involves putting him in modern clothes. The politics remain the same.

Macbeth by William Shakespeare is set in a quite different culture than the contemporary western one. The characters are formed by their environment and the plot constructed around incidents and belief systems quite unlike modern western ones. Thus you cannot actually "modernize" the character of Macbeth. You could simply invent another character with a limited number of similarities, but it would not be a modern Macbeth, simply a different character.

If you wanted a 21st century character who could murder a king to become an absolute monarch, you could set the play in the Middle East or sub-Saharan Africa, where a few absolute monarchies still exist, but there might be some difficulty in transposing other elements of the play such as the role of Lady Macbeth (the majority of 21st century absolute monarchies are also polygamous). Witches would fit in an animist setting in Africa, but not in an Islamic one. 

There is no reason, however, that the character of Macbeth needs to be changed to "suit" 21st century audiences. Part of the point of experiencing literature is to learn about cultures of other times and places. Modernizing earlier works means that instead of learning about different cultures, you just end up encasing yourself in the narrow bubble of your own culture.