Here's a guess. $K_2(\mathbb{C})$ is described in terms of Steinberg symbols as the quotient of $\Lambda^2_\mathbb{Z}(\mathbb{C}^*)$ by the elements $z\wedge(1-z)$. My guess is that the map sends whatever's left (if anything) of $H_1(\mathbb{C}^*,\mathbb{C})$ to zero, and sends $E_2^{2,0}=\Lambda^2_{\mathbb{Z}}(\mathbb{C}^*)$ surjectively onto $K_2(\mathbb{C})$ via the Steinberg symbol map. You might be able to extract a proof from the paper of Dupont and Sah, "Scissors congruences, II". 

In fact, (A11) of Dupont & Sah says that the map from the upper triangular matrices to $\mathbb{C}^*$ with kernel $\mathbb{C}$ is a homology isomorphism, so there's nothing left of $H_1(\mathbb{C}^*,\mathbb{C})$, and $H_2(G,\mathbb{Z})$ is just $\Lambda^2_{\mathbb{Z}}(\mathbb{C}^*)$.