There is a homological Chern character $ch_\ast \colon K_\ast(X) \to H_\ast(X)$ for $X$ a smooth, compact manifold.

I found only one definition of it (in the paper "K-Homology and Index Theory" by Baum, Douglas), where it is defined (i) by the usual cohomological Chern character map and Poincare duality between $H^\ast(X)$ and $H_\ast(X)$, and (ii) by using the geometric definition of the K-homology groups $K_\ast(X)$ of $X$.

(The geometric definition of K-homology uses cycles $(M, E, \phi)$, where $M$ is a closed Spin$^c$ manifold, $E$ a complex vector bundle on it and $\phi \colon M \to X$ a continuous map. Then $ch_\ast (M, E, \phi) := \phi_\ast(ch^\ast(E) \cup Td(TM) \cap [M])$, where $ch^\ast$ is the cohomological Chern character.)

Especially the use of Poincare duality to reduce the definition to the usual cohomological Chern character bothers me.

Is there some direct definition of the homological Chern character?

With "direct" I mean not using Poincare duality and then the cohomological Chern character. It would also be nice if this direct definition would not use the geometric description of K-homology but either the one by Fredholm modules or through the dual algebra $\mathfrak{D}(C_0(X))$.

Maybe one has to go through cyclic (co-)homology or Hochschild homology, e.g., first define $K_\ast(X) \to HC$ and then $HC \to H_\ast(X)$. That would be ok for me.