According to this set of notes, which says (paraphrasing):
"To construct the homotopy fixed points spectral sequence, we use the fact that the bar construction gives a simplicial resolution of $(EG)_+$ which yields a cosimplicial object etc."
I believe what is meant is that we consider the sets, viewed as discrete topological spaces: $$ B^0_+ = B^0(G,G,*)_+ \Leftarrow B^1_+ = B^1(G,G,*) \Lleftarrow\cdots $$ Apply $F(-,X)^G$ $$ F(B^0,X)^G_+ \Rightarrow F(B^1_+,X) \Rrightarrow_ \cdots $$ and this is literally the cosimplicial object we feed into the Bousfield-Kan spectral sequence.
Is this correct, or is there something basic I'm missing?