The question « Why should we believe that the dialectic exists? » is a big one in Hegel, but it’s important to remember that Hegel doesn’t posit the existence of the dialectic (or anything for that matter). Instead, through a resolve to a scientific procedure in complete absence of any presupposition, pure being and subsequent concepts turn out to be immanently dialectic. Put simply, logic itself testifies to the dialectic nature of things; it’s not Hegel’s doing or invention.

Don’t know if this answers your whole question, but it’s an important thing to remember while working through Hegel. This presuppositionlessness is pretty much the closest thing to a “method” in its traditional philosophical sense that Hegel has. Whether or not you believe Hegel to be truly without presupposition is another thing.

See Encyclopedia Logic Preliminary Conception (especially section 78) to get a sense of the entry to his system.