What is the oxidation state of C in this lewis structure?
This answer uses electronegativities for the calculation of oxidation states as proposed in the Expanded Definition of the Oxidation State by Hans-Peter Loock in 2011.

Foreword: As mentioned in the OPs comments, electronegativity and oxidation states are atom properties, not group properties.

Comparing Electronegativities

This table shows an excerpt from Pauling electronegativities ($\chi_{\mathrm{Pauling}}$):

$$\begin{array}{cc}
\hline
\text{Element} &amp; \chi_{\mathrm{Pauling}}\\
\hline
\ce{H} &amp; 2.20\\
\ce{C} &amp; 2.55\\
\ce{O} &amp; 3.44\\
\hline
\end{array}$$

Hence, the following applies:

$$\chi_{\mathrm{Pauling}}(\ce{H}) &lt; \chi_{\mathrm{Pauling}}(\ce{C}) &lt; \chi_{\mathrm{Pauling}}(\ce{O})$$

Looking at heteroatomic Bonds ($\ce{A-B}$, $\ce{A=B}$, and $\ce{A#B}$)

Since the oxygen atom has a higher electronegativity than the carbon atom, it attracts the bonds atoms more, resulting in an polarized bond. Presuming one would apply pulling force to the two atoms in a $\ce{C-O}$ or $\ce{C=O}$ bond until the atoms separate, it is assumed that the bond cleavage is heterolytic, thus allocating the bonding electrons to the oxygen atom.

The same applies to $\ce{C-H}$, $\ce{O-H}$, or any other heteroatomic bond ($\ce{A-B}$, $\ce{A=B}$, and $\ce{A#B}$).



Putting the jigsaw puzzle together

Now, having a look on the complete structure of the molecule in question, and applying the aforementioned rule to each bond:



Last thing to do is calculating the atoms hypothetical charge after separation, which is to be equatable with the oxidation state:

$$ \text{Oxidation state} = N_\mathrm{i}(\ce{e-}) - N_\mathrm{f}(\ce{e-}) $$

With $ N_\mathrm{i}(\ce{e-}) $ representating the number of electrons in a free atom, and $ N_\mathrm{f}(\ce{e-}) $ the one after separation (Never forget the lone pairs).

This will lead to the following oxidation states for each atom:



$$\begin{array}{cccc}
\hline
\text{Atom} &amp; N_\mathrm{i}(\ce{e-}) &amp; N_\mathrm{f}(\ce{e-}) &amp;\text{Oxidation state}\\
\hline
\ce{H-1} &amp; 1 &amp; 0 &amp; +1\\
\ce{H-5} &amp; 1 &amp; 0 &amp; +1\\
\hline
\ce{C-2} &amp; 4 &amp; 2 &amp; +2\\
\hline
\ce{O-3} &amp; 6 &amp; 8 &amp; -2\\
\ce{O-4} &amp; 6 &amp; 8 &amp; -2\\
\hline
\end{array}$$

The structure with assigned oxidation states:



The oxidation state of the carbon is $\mathrm{+II}$.