My take on it would be that if the Klingon Empire wanted to become a Federation Member State, a number of changes would need to take place.
First, the Empire would need to adopt some variant of liberal democracy. Given the resemblance between the Empire's government and the British government of the past, I imagine that something akin to a Westminster-style system would work best for them: A democratically-elected High Council, with the Chancellor being appointed by the Emperor from among the High Council on the basis of which leader commands the support of a majority of Council members.
The next question would be what to do about the Klingons' conquered worlds. I imagine that the Federation would require, as a condition of Membership, that each world be allowed to hold a referendum on either continued membership within the Empire (with their citizens being entitled to democratic representation on the High Council) or independence, with the results of each referendum being respected.
The third condition, I imagine, would be the abolition of the caste-based discrimination that is known to occur within Klingon society ("Once More Unto the Breach").
Once all those factors have been met, I suspect that the Federation would consider accepting the Klingon Empire as a Member State. The question, though, would become whether or not the Empire should be accepted given its sheer size. If the Federation Councillors from each Member State are determined on the basis of the population size of that Member (akin to the U.S. House of Representatives), that could lead to the Federation Councillors from the Klingon Empire being able to dominate the Federation Council unfairly. On the other hand, if each Federation Member State gets an equal number of Federation Councillors (as the novels tend to depict, similar to the U.S. Senate), that, too, would be unfair -- a Klingon Federate's vote would be worth considerably less than a Vulcan Federate's, for instance, because there would be so many more Klingon Federates. Either way, I suspect that the Federation may decide that the Empire needs to be broken up into smaller component states before it could be granted Membership.
Alternately, the Federation may decide that it will simply not accept the Klingon Empire as a Federation Member unless it immediately grants independence to all of their conquered planets, period, as a way of avoiding the need to break it up into multiple Klingon-dominated Federation Member States.