In the American English sense of "person liberally and excessively sympathetic" (especially toward those the speaker or writer deems not to deserve it) is attested by 1936 in the work of popular conservative newspaper columnist Westbrook Pegler (1894-1969), who first used it in reference to his own feelings about the Republican Party but by 1938 regularly deployed it against the Roosevelt administration and also as a modifier (bleeding-heart liberal) in his "Fair Enough" column:
And I question the humanitarianism of any professional or semi-pro bleeding heart who clamors that not a single person must be allowed to hunger, but would stall the entire legislative program in a fight to jam through a law intended, at the most optimistic figure, to save 14 lives a year. ["Fair Enough," in Freemont (Ohio) Messenger, Jan. 8, 1938]
Bleeding in a figurative sense of "generous" is attested from late 16c., and the notion of one's heart bleeding as a figure of emotional anguish is from late 14c.; the exact image here may be the "bleeding heart of Jesus."