While it's unlikely that Trump's claim that Obama ordered a wiretap on the Trump team in the Trump Tower before or after the election, there may be something to the claim that the US Government was listening in to conversations by Trump people in the tower while Obama was still President.
Former National Security Adviser Flynn had to resign because of intercepted telephone conversations while Trump was President-elect, according to the Washington Post. So, somebody, for some reason, was listening to Flynn's phone calls while he was working on the team of President-elect Trump.
It seems most likely that the National Security Agency listens in on all calls involving Russian diplomats, although if both ends of the calls are entirely with the United States, then the FBI might be listening, rather than NSA. I'm not sure who has the lead in this case. The niceties of whether the call was listened to because it involved the Russians rather than Trump officials may be lost on Trump, who considers any listening in on his teams' conversations a violation of privacy, i.e., a violation of the Fourth Amendment.
The NSA probably routinely picks up many phone calls involving Americans in its huge net of intercepted phone calls. It had no qualms about intercepting Angela Merkel's phone calls in Germany. However, all such conversations involving American citizens inside the US are supposed to be highly protected because of the Fourth Amendment. In this case, these phone calls involving a American citizen, Flynn, were leaked to the press, a violation of the laws dealing with classified information as well as the Fourth Amendment.
It's not surprising that Trump is mad. He is wrong that Obama ordered a wiretap, but he is correct that his people were tapped by the US Government while Obama was still President, albeit under procedures that had been in place for years.
A more relevant question might be whether Obama had any knowledge of the leaks about Flynn and looked the other way. Was this leak really done by somebody in the "deep state" devoted to Hillary Clinton as Obama's successor? If so, was the Obama administration culpable by looking the other way while such leaks were going on? The leak was a violation of law, but like the misdeeds of the bankers in 2008, no one is being investigated because of the leak.