The persuasive techniques employed by Socrates can best be analyzed in terms of ancient Greek rhetorical theory. Many extant Greek texts, including the Rhetorica ad Alexandrum and Aristotle's Art of Rhetoric analyze various forms of rhetorical appeal.
In looking at the persuasive strategies of Socrates, first you should ask yourself of what Socrates is actually trying to persuade an audience. Is he trying to convince them that death is better than life? Or is this what was known as a "figured problem" in ancient rhetorical theory, in which his overt arguments concerning the positive aspects of death were actually being deployed to advance covert arguments concerning how he wished the Athenians to act? This type of analysis, determining the "issue" or stasis around which a case resolved was the first step of rhetorical analysis.
Next, you might examin the three classical rhetoric proofs from ethos, logos and pathos. How does Socrates project his own character? How does his calm and bravery serve as a persuasive device? Next, is his refusal to engage in pathetic appeal, itself an appeal to a different type of emotion? Finally, what logical arguments does he make?
Also, you should look for standard rhetorical topoi such as his comparison and contrast of life and death, his definition of death, etc.