Let $X_n$ denote the n-th skeleton. The CW chain complex of $S^1 \times S^1$ is just $$...0 \rightarrow (X_{-1} = \emptyset) \rightarrow (X_0 = \bullet) \rightarrow (X_1 = S_1 \vee S_1) \rightarrow (X_2 = S^1 \times S^1) \rightarrow 0 ... $$
So attach one $0$-cell, two $1$-cells, and one $2$-cell.
We can generalize this to say that (for example) the chain complex for $S^2 \times S^2$ is $$...0 \rightarrow (X_{-1} = \emptyset) \rightarrow (X_0 = \bullet) \rightarrow (X_1 = \bullet) \rightarrow (X_2 = S^2 \vee S^2) \rightarrow (X_3 = S^2 \times S^2) \rightarrow 0 ... $$
Is that correct? I just wanted to be sure...