How does the author reveal Hamlet’s fragile state, despite slips from reality?

Your question concerning Shakespeare's
Hamlet is a bit confusing.  You use the word "despite" even though
breaks from reality would definitely reveal a "fragile state."  I don't know that Hamlet
suffers any breaks from reality, but since you don't want those anyway, I'll deal with
other revelations. 

Hamlet's fragile state is revealed
mostly during his speeches.  In Act I he classifies the world as an "unweeded garden." 
Hamlet is taking his own situation (unexpected loss of a father, hasty remarriage of his
mother, and loss of the crown that he must have considered his--these are revelations of
his shaky state, too, by the way) and applying it to the state of the world and of human
existence. 

Hamlet also wishes at one point that God had
not declared suicide a sin.  Obviously, he is
contemplating it.

In the famous "To be or not to be"
speech, Hamlet questions the value of existence. 

Hamlet
sees all women as unfaithful, lustful, frail, ungrateful, once again applying his own
situation (his problems with his mother and Ophelia) to the world as a whole. 
 

Hamlet also uses sarcasm and satire in his conversations
with others (Claudius, Polonius, Gertrude, Ros. and Guil.), possibly a sign of
depression.

All in all, Hamlet is depressed and
despairing.  He definitely is in a fragile state.