Explanation for the Chern character

For a general complex oriented cohomology theory represented by a ring spectrum $R$, there is a "Hurewicz map" from $R$ to its smash product $H\mathbb{Z}\wedge R$ with the Eilenberg-Mac Lane object for the integers. $R$ has a formal group law associated to it as you stated. So does $H\mathbb{Z}\wedge R$; in fact, it carries the formal group law from $H\mathbb{Z}$ (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 $\pi_*R$ to $H_*R$ is the universal map from $\pi_*R$ (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.

There is a beautiful explanation of the Chern character that my student Fei Han proved in his thesis: The Chern character is given by the map that "crosses with the circle". The hard part is to explain why domain and range of this map are K-theory respectively de Rham cohomology. This uses isomorphisms

$K^0(X) \cong 1|1-EFT[X]$ and $H^{ev}_{dR}(X) \cong 0|1-EFT[X]$

where $d|1-EFT[X]$ are concordance classes of $d|1$-dimensional Euclidean field theories over the manifold $X$. Since the circle of length one is a Euclidean 1-manifold, it is not hard to believe, modulo the precise definitions, that crossing with it gives a map as required.

In fact, his result works even before taking concordance classes, where the left hand side is replace by vector bundles with connection and the right hand side becomes (even closed) differential forms.

There is a nice discussion about multiplicative sequences, &c., in Lawson and Michelsohn's book "Spin Geometry". It discusses things like the Todd genus, the A-hat genus, and so on, but also the Chern character and the ring homomorphism from K-theory to ordinary cohomology. It is a readable exposition and perhaps "connects the dots" in a way that would be helpful to you.