In "Hamlet" how do Ophelia's actions and apparel indicate she is not well?
First of all, she is talking nonsensically, and singing nursery rhymes and songs. For example, she goes from singing about flowers and graves to telling the king,
"They say the owl was a baker's daughter. Lord, we know what we are, but know not what we may be. God be at your table!" (IV.v.42-44)
She is jumping all over the place in her speaking, and most of it does not make any sense. As the Queen and Horatio speak of her, they are very concerned. They say that she "speaks much of her father...hems, and beats her heart...her speech is nothing" (IV.v.4-7). They say that as she speaks, you think it is nonsense, but she seems so sincere, and her gestures, "her winks, and nods, and gestures" make the listener try to understand (IV.v.11). It is almost as if she has reverted into childlike behaviors-talking nonsense in a most sincere way, singing nursery songs, asking questions and giving reassurances based on illogical conclusions. It could be she has reverted to these things in order to better cope; she can't handle reality, so she has gone back to childhood when things were happier.
In the actual text of the play Shakespeare doesn't give much to indicate her appearance or dress; if you watch the movies of the play, or go to a play, most have her looking disshevelled and unkempt, wearing loose-fitting, childlike clothing. For a lady of the castle, that is completely inappropriate and quite startling. So, her appearances as interpreted by directors, and her actual words do leave us worried for her, and wondering about her sanity.