The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define the Chern character on a line bundle as $\mathrm{ch}(L) = \exp(c_1(L))$ and then extend this; then for example $\mathrm{ch}(L_1 \otimes L_2) = \exp(c_1(L_1 \otimes L_2)) = \exp(c_1(L_1) + c_2(L_2)) = \mathrm{ch}(L_1) \mathrm{ch}(L_2)$; then we can use this to define a Chern character on general vector bundles.
This all seems a bit ad-hoc, and it doesn't give much insight as to why such a thing exists anyway.
An explanation I like a lot better comes from even complex oriented cohomology theories. Given any complex oriented periodic cohomology theory, such as K-theory or periodic (ordinary) cohomology, we have $H(\mathbb{CP}^\infty) \cong H(P)[[t]]$ for $P$ a point. Seeing as $\mathbb{CP}^\infty$ is the classifying space for line bundles, this gives us a way of having "generalised Chern classes" for any line bundle corresponding to any cohomology theory, and even for any vector bundle.
We have a link between complex oriented periodic cohomology theories and formal group laws, corresponding to what corresponds to $c_1(L_1 \otimes L_2)$ in $\mathbb{CP}^\infty$: for ordinary cohomology, as above, we get that $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2)$ which gives the additive formal group law, and for K-theory we get $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2) + c_1(L_1) c_2(L_2)$ which is the multiplicative formal group law. The fact that over $\mathbb{Q}$ (but not over $\mathbb{Z}$) there is an isomorphism between the formal group laws given by the exponential map, and this reflects in the cohomology, giving the Chern character $K(X) \otimes_\mathbb{Z} \mathbb{Q} \to \prod_n H^{2n}(X,\mathbb{Q})$.
I'm not too sure what the exact formulation in that second case is, but more importantly I was wondering if there are any other, cleaner interpretations of the Chern character (I've been hearing about generalised Chern characters, and I have no idea where they would come from in this case). It seems like there should be a way to link the Chern character to things like the genus of a multiplicative sequence, and tie it in with other similar ideas for example the Todd genus or the L-genus given by similar formal power series. I guess the trouble is that I don't see how these related ideas all fit in together.
Best Answer
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.