If $C$ is a graded coalgebra (e.g. $C$= the homology of a d.g. Hopf algebra), then $C_0$ is not necesarily a subcoalgebra, because  
$$\Delta(C_0)\subset (C\otimes C)_0=\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$$

For example, $H=k\{x,y\}/(x^2=y^2=xy+yx)$ is a graded Hopf algebra with $|x|=1$ and $|y|=-1$, both $x$ and $y$ primitives  (in particular a d.g. Hopf algebra with $d=0$ and agree with its homology).

$H_0=k\oplus kx\wedge y$, and
$$
\Delta(x\wedge y)=
(x\otimes 1+1\otimes x)(y\otimes 1+1\otimes y)$$
$$=
x\wedge y\otimes 1+1\otimes x\wedge y+ x\otimes y-y\otimes x
\notin H_0\otimes H_0$$

Of course if $C=\oplus_{n\geq 0} C_n$, then $C_n=0$ for $n<0$ and $C_{-n}=0$ for $n>0$, so, the only nonzero summand in $\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$ is $C_0\otimes C_0$.