Shakespeare’s plays, both his tragedies and his histories, have been examined with reference to the Seven Deadly Sins—jealousy (Othello) being the play most often cited because so obvious. Not exactly the same as a “tragic flaw” but comparable, they enter into the otherwise idealized world of the characters in the form of “imperfections of character," bringing unwanted circumstances to the plot. Take, for example, Falstaff’s gluttony in Henry IV; while he is not the title character, his indulgences certainly taint Henry’s youth, and threaten to ruin his ascent to the throne until Henry (no longer Hal) disowns him—“I know thee not, old man.”
Greed is ubiquitous in the canon—Lear’s daughters, for example. So one answer to how evil inserts itself into Shakespeare’s plays is via human nature’s imperfections. It’s true that Shakespeare occasionally depicts an absolutely evil character (for example, perhaps Lady Macbeth, or Caliban), but they are few. The reason is that Shakespeare, more than any other playwright, made use of the innovative device of “character”—that is, real human beings depicting all the psychological traits of real people. It is hard to realize that they didn’t exist to any degree in literature before the Renaissance, but Humanism (14th-15th centuries) was the intellectual movement turning art from religious subjects to the legitimate study of Man. When Shakespeare introduced the concept of “characters” on stage, “the evil that man do” came with them.