In the first pass, temporarily relabel all samples in class 'two' and 'three' to (both) be numbered #4, leaving class 'five' as class #5. Train on that and save the resulting classifier ('A').
In the second pass, remove all samples in class #5, and train 'two' #2 vs 'three' #3, and save the resulting classifier ('B')
Now, to use this, classify the unlabeled samples using classifier A. The samples will get classified as being in class #5 or in #4. Extract the ones classified #5: those belong to class 'five'.
Take everything that was left (#4), remove the #4 label, and classify the (now unlabeled) samples using classifier B. The samples will get classified as being in class #2 ('two') or in #3 ('three').
Now, each sample was classified as 'five' by running the first classifier, or was classified as 'two' or 'three' using the second classifier. The task is finished.
To generalize: create a classifier which distinguishes between the first class, and (everything in all other classes). Then remove all samples from the first class from consideration, and create a classifier that distinguishes between the second class and (everything that is not in the first or second class). Continue on.
In the N'th step, remove all samples from the first (N-1) classes from consideration and create a classifier that distinguishes the N'th class from (everything except the first N classes). Keep doing that until your final classifier is between the second last class and the last class.
Once you have created all of those classifiers, apply them in sequence on the unknown data. In step #N of this, if the N'th classifier classes a sample as being in the first of its two classes, then those samples belong to class #N, so remove those from consideration and keep going, applying classifier #(N+1) to what remains, until eventually all samples have been extracted or you have reached the output of the last classifier (in which cases its second class correspond to the last class of the original list of classes.)
