  In this paper we construct a bivariant Chern character defined on ``families
of spectral triples''. Such families should be viewed as a version of unbounded
Kasparov bimodules adapted to the category of bornological algebras. The Chern
character then takes its values in the bivariant entire cyclic cohomology of
Meyer. The basic idea is to work within Quillen's algebra cochains formalism,
and construct the Chern character from the exponential of the curvature of a
superconnection, leading to a heat kernel regularization of traces. The
obtained formula is a bivariant generalization of the JLO cocycle.
