The existing answer from Deusovi is good, but it can also be solved in a way that will, one third of the time, only need 2 questions, and the rest of the time still be solved with a third question. I'm not saying that's better, but it might be interesting anyway...
Please excuse my formatting troubles, it's my first answer on here.
Try to find the coin-flipper earlier
Ask a question to which the true answer is yes. To ensure this is something we know that they all know we can use what we've been told in the statement of the problem - that the 5 of them know how the others answer and we can ask "Does one of you always lie?"
I've assumed that we can vary the follow up questions based on the answer before.
The first question will have been answered yes by truth-teller and always-says-yes and answered no by liar and always-says-no, so if the coin-flipper says yes there will be 3 people saying yes, and if they say no there will be 3 saying no. Randomly select one of the 3 that gave the same answer ask everyone "Is this the coin-flipper?" (i.e. if 3 said yes select someone who said yes, if 3 said no select someone who said no)
If the coin-flipper is the one selected then the truthful answer to question 2 is again yes, so in that case all 4 of the non-coin-flippers answers will all stay the same as question 1
If the selected person is not the coin-flipper then the truth-teller and the liar (and maybe the coin-flipper) will change their answers.
So if 2 or more change their answers, the coin-flipper is the selected person, otherwise they are not.
If the coin-flipper was the selected one then this requires a 3rd question to identify the other 4 people, anything for which the true answer is no will do, for example "Do all of you always tell the truth?" The liar will now have answered no,no,yes; the truth-teller yes,yes,no and the always-no and always-yes will be obvious.
The remaining cases are where we know the selected person is not the coin-flipper. If 3 people said yes to the first question: Answers of No,Yes implies liar; No,No -> always-no; Yes,No -> truth-teller or coin-flipper; Yes,Yes Always-Yes or coin-flipper. The coin-flippers answers will match another persons, but if they match the selected person's then, because we know the selected person is not the coin-flipper we can identify everyone without asking a 3rd question. If they do not match the selected person, then a third question is required to distinguish between the coin-flipper and the other (non-selected) person with the same answers. Pick either of these 2, ask "is this the coin-flipper?", since the liar has been identified we can use their incorrect answer to identify the last 2.
The logic is similar, with some inversions for the case where 3 people said no to the first question and the selected person is not the coin-fipper: Answers of Yes,Yes -> Always-says-yes; Yes,No -> Truth-teller; No,No -> Always-says-no or coin-flipper; No,Yes -> Liar or coin-flipper. Again if the coin-flippers answer matches the selected person then no further question is required, otherwise ask the same question noting that in this case the truth-teller has already been identified.
Why this needs 2 questions one third of the time:
The cases that only needed 2 questions require that the person selected to ask about in question 2 is not the coin-flipper, There is a 2/3 chance of that. It also requires that the coin-flippers random answer to that question matches the answer given by the selected person 1/2 chance of that. 2/3 * 1/2 = 1/3 chance of only needing 2 questions.