I would recommend doing this stably with spectra, at least to start with. A key ingredient is this theorem of Reg Wood: there is an equivalence $KO/\eta\simeq KU$, where $\eta\in\pi_1KO=\widetilde{KO}^0(\mathbb{R}P^1)$ corresponds to the tautological bundle minus one. (In the comments Qiaochu Yuan refers to a "mysterious map" $BO\to O$; this is essentially $V\mapsto\eta\otimes V$.) This gives a cofibre sequence $$ \Sigma KO \xrightarrow{\eta} KO \xrightarrow{f} KU \xrightarrow{g} \Sigma^2KO. $$ Here $f$ is the obvious complexification map. To describe $g$, let $\nu\in\pi_2KU$ be the usual generator (which is invertible), and let $h\colon KU\to KO$ denote the forgetful map; then it can be shown that $h=g\circ\nu^{-1}$. This can now be twisted around to give a cofibre sequence $$ \Sigma^{-2}KO \xrightarrow{\nu^{-1}\circ f} KU \xrightarrow{h} KO \xrightarrow{\eta} \Sigma^{-1}KO, $$ which gives information about the image of $h$, as required. The zeroth spaces in the above sequence are $$ O/U \to \mathbb{Z}\times BU \to \mathbb{Z}\times BO \to O. $$ The first map is trivial in mod 2 (co)homology. If we just look at the base components, the other three spaces give a diagram of Hopf algebras $$ \mathbb{F}_2[c_k|k>0] \xleftarrow{} \mathbb{F}_2[w_k|k>0] \xleftarrow{} \mathbb{F}_2[w_{2k-1}|k>0] $$ The first map sends $w_{2k}$ to $c_k$ and $w_{2k-1}$ to $0$; the second map is the obvious inclusion.

All of these things are covered in an integrated way in my thesis using Hopf rings, although of course all individual pieces of the story are much older.

I would recommend doing this stably with spectra, at least to start with. A key ingredient is this theorem of Reg Wood: there is an equivalence $KO/\eta\simeq KU$, where $\eta\in\pi_1KO=\widetilde{KO}^0(\mathbb{R}P^1)$ corresponds to the tautological bundle minus one. (In the comments Qiaochu Yuan refers to a "mysterious map" $BO\to O$; this is essentially $V\mapsto\eta\otimes V$.) This gives a cofibre sequence $$ \Sigma KO \xrightarrow{\eta} KO \xrightarrow{f} KU \xrightarrow{g} \Sigma^2KO. $$ Here $f$ is the obvious complexification map. To describe $g$, let $\nu\in\pi_2KU$ be the usual generator (which is invertible), and let $h\colon KU\to KO$ denote the forgetful map; then it can be shown that $g=h\circ\nu^{-1}$. This can now be twisted around to give a cofibre sequence $$ \Sigma^{-2}KO \xrightarrow{\nu^{-1}\circ f} KU \xrightarrow{h} KO \xrightarrow{\eta} \Sigma^{-1}KO, $$ which gives information about the image of $h$, as required. The zeroth spaces in the above sequence are $$ O/U \to \mathbb{Z}\times BU \to \mathbb{Z}\times BO \to O. $$ The first map is trivial in mod 2 (co)homology. If we just look at the base components, the other three spaces give a diagram of Hopf algebras $$ \mathbb{F}_2[c_k|k>0] \xleftarrow{} \mathbb{F}_2[w_k|k>0] \xleftarrow{} \mathbb{F}_2[w_{2k-1}|k>0] $$ The first map sends $w_{2k}$ to $c_k$ and $w_{2k-1}$ to $0$; the second map is the obvious inclusion.

All of these things are covered in an integrated way in my thesis using Hopf rings, although of course all individual pieces of the story are much older.