First of all, it is extremely difficult to know anything about the historical Socrates. So, I will just be referring to Socrates as he appears in Xenophon.

Was there any overlap in philosophy/ideals?

Yes. All of them were elitist, anti-democratic admirers of Sparta. In fact, Sparta installed the Thirty as a satellite government of Athens after Sparta defeated Athens in the Pelopennesian War.

Can he be considered complicit in their actions to any degree?

It is difficult to say. Socrates probably did exert some influence over some members of the Thirty when they were younger, but it is difficult to know whether that influence caused any of them to act the way they did when they were in power. Xenophon--a devoted associate of Socrates--argues that Socrates was not responsible. If you would like more information, see I. F. Stone's The Trial of Socrates.

How much of a factor was association with them in charges being brought against Socrates after their reign ended?

According to Xenophon, it was a big factor. One of the charges against Socrates was that Socrates corrupted the youth. Xenophon writes that Socrates' "accuser" (presumably Meletus) argued that Socrates' association with Critias was evidence for the charge. Critias was a leader of the Thirty and one of the most bloodthirsty of them. Socrates' accuser was claiming that Socrates was a cause of Critias' future behavior.

Furthermore, Anytus was one of Socrates' accusers. Anytus was a democrat who helped lead the forces that defeated the Thirty. Anytus' action against Socrates may have been motivated by resentment over Socrates' association with the Thirty.

However, it is important to note that there are other factors involved. For example, Socrates had had an association with Anytus' son, and Anytus may have resented Socrates' influence on his son.

Xenophon claims that Socrates criticized the Thirty's behavior. Similarly, in Plato's Apology, Socrates says that he refused to obey an order that the Thirty issued him. In fact, Xenophon says that Critias bore a grudge against Socrates and that Critias outlawed teaching the art of words as a calculated insult against Socrates. Socrates' exact association with the Thirty is unclear.

You might be interested in my answer to a closely related question:

Did the Athenians really believe that Sokrates was corrupting their youth?

In my view, it was largely his association with the Thirty (whether active or not) that drove the Athenians to execute him.