For a general complex oriented cohomology theories theory represented by a ring spectrum R, there is a "Hurewicz map" from R to its smash product HZ ∧ R with the Eilenberg-Maclane Eilenberg-Mac Lane object for the integers. R has a formal group law associated to it as you stated. So does HZ ∧ R; in fact, it carries the formal group law from HZ (the additive group), the one from R, and an isomorphism between them. You can think of this isomorphism of as a "logarithm" for the formal group law of R.
For certain complex oriented cohomology theories R (the so-called "Landweber exact" theories) you can say more. Complex K-theory, which is represented in the stable homotopy category by a spectrum called KU, is one such example. In Landweber exact cases, the Hurewicz map of graded rings from π
*R to H
*R is the universal map from π
*R (with its formal group law) to a ring where this formal group law has a choice of logarithm.
In the case of K-theory (and in some other cases), this universal ring is the rationalization. So you can think of the Chern character as simply the Hurewicz homomorphism, or as the universal way to adjoin a logarithm to the formal group law of K-theory.