This question is driving me crazy - it was asked on a bulletin board
about books and no one has yet been able to asnwer it.  I have looked
on numerous sites for help on finding a book I read years ago.  A boy
meets these two old ladies that are living in poverty.  He brings them
food and helps them around the house.  One lady accepts his help, the
other rejects it.  They both quote Shakespeare a lot and he learns it
so he can talk to them, it helps him do well in English class but his
other subject's suffer because he's working to try and support them. 
Eventually he finds out the two ladies are actually the same person,
the woman has dual personalities or is having fun with him.  The lady
dies in the end and leaves a fortune to the boy.

Hello. 

Your book was likely "The Dream Watcher" (1968) by Barbara Wersba.

From loganberrybooks.com:

" The main character, the boy, doesn't fit in at home or school... He
meets an old woman who is manifestly eccentric, and claims to be a
retired actress, who once knew Sarah Bernhardt. She helps the boy
learn to be himself. Its only when she dies that the boy finds out she
was nuts.. she was making up her entire life, whole-cloth."
http://loganberrybooks.com/solved-d.html


From The New York Times review:

"The Dream Watcher is the story of Albert Scully, a high-school boy...
It is also the story of Orpha Woodfin, a marvelous old nut...
Miss Wersba has bravely undertaken the difficult stylistic
accouterment of much quoted material from Shakespeare...
The old lady, living in the only remaining  original house in the
midst of a new development, bills herself as a bygone actress who once
played Juliet..."
source:
The New York Times, November 3, 1968, p. 469
http://pqasb.pqarchiver.com/nytimes/results.html?st=advanced&QryTxt=%22dream+watcher%22&x=0&y=0&By=weston&Title=&datetype=6&frommonth=11&fromday=03&fromyear=1968&tomonth=11&today=03&toyear=1968&restrict=articles&sortby=REVERSE_CHRON
 
Vintage copies available from Abebooks.com starting at $1.94.
http://www.abebooks.com/servlet/SearchResults?bx=off&sts=t&ds=30&bi=0&an=wersba&y=0&yrl=1968&tn=dream+watcher&x=0&sortby=2&yrh=1974

-------
search strategy:
abebooks, advanced search terms: old lady fiction (1965-1975)

I hope this helps.

Thanks so much!