I think the comment from Marek Oleszczuk got the keyword, here are my contributions for the other lines:
1/ You might think I'm broke, but then
Lady in Red by Chris de Burgh. If you are "in the red", financially, you have a negative balance, and being broke is a similar state.
2a/ I'm a singer...
Lady Madonna by the Beatles. Madonna is of course, among her other talents, a popular singer.
2b/ ...and sound like a hen
Lay Lady Lay by Bob Dylan (courtesy of @TroyAndAbed). Hens lay eggs.
3/ You can count on me
Three times a Lady by the Commodores. The line from the song containing the title is "you're once, twice, three times a lady", hence the counting.
4/ I'm a tramp you see
The Lady is a Tramp by Frank Sinatra (taken from comments). This is self-explanatory, I think.
5/ So I'll sing you the blues once again.
Lady sings the blues by Billie Holiday. Again this is self-explanatory.
The key word is of course
Lady. Each clue is told from the point of view of the particular Lady as designated in the song title, although sometimes interpreted differently (cryptically even) to the original.