A similarity between Dr. Jekyll and Macbeth is that both these men try their best to keep the evil sides of their characters hidden from other people. Dr. Jekyll presents the persona of a benevolent doctor and a conservative English gentleman, while Macbeth presents the persona of a noble aristocrat and a loyal subject. In both cases these men venture into evil knowing it is evil but thinking that they can draw back into their "good" personas whenever they want to. In Jekyll's case the withdrawal would come when he had finished his experiments, while in Macbeth's case the withdrawal would come when he had gotten the crown he wanted. But in both cases they find that one step into evil leads to another and another. Macbeth says:
I am in blood
Stepp'd in so far, that, should I wade no more,
Returning were as tedious as go o'er.
Dr. Jekyll finds (as people often do when playing with illicit drugs) that he needs a stronger and stronger doseage, and eventually he loses control of himself altogether.
Basically, Dr Jekyll and mr Hyde is about split personality. Macbeth is nicknamed "Black Macbeth" and incarnates evil in the story, "smacking of every sin that has a name". Malcolm, his successor on the throne represents good.The alliterative names Macbeth, Macduff and Malcolm point to the potential relation between the dispersed parts of a single unique self, that of man, torn between the forces of good and evil, just as in Dr Faustus.
The similar phonic characteristics between those names calls to mind the names of Richmond and Richard in Richard III. They are enemy brothers. The triad of the "sons" or "suns" of York is the corollary of the three heroes in Macbeth. The odd thing is that Shakespeare "blackened" Macbeth but Malcolm, in Act IV, scene 1 opposes the the alleged tyranny of his own nature to Macbeth's bloody tyrannical power and to a certain extent, "whitewashes him". This is probably due to the fact that even he can be influenced by Macbeth's baneful personality.
Doctor Jekyll or "Je-kill" and Mr "hide" can be associated with Macbeth's rhetoric of desire and , the idea of scientific creation substituted for witchcraft. But the main difference resides in the different ending since in Macbeth, the better self, Malcolm, survives and wins over Macbeth whereas in Stevenson's 19th century novel, the sacrificial crisis destroys both symbolic twins.
We can compare both Macbeth and Jekyll & Hyde to S. Freud's ideology in that Lady Macbeth and Hyde are the I.D (the devil on your shoulder) whereas I would say Jekyll is both Ego and Superego and in Macbeth, Banquo is the Superego, Macbeth being the Ego. In Macbeth, the problem or situation posed is the witches and their prophecy.
Sorry if this answer isn't quite as expected, but I'm not yet 15, and am still doing my CAT. Therefore my use of language may be a tad stiff, but the above paragraph was part of a homework draft.