Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of for coalgebras over the sphere spectrum. We construct a relative coBökstedt spectral sequence to study of coalgebra spectra over any commutative ring spectrum R. Further, we use algebraic structures in this spectral sequence to complete some calculations of the homotopy groups of relative topological coHochschild homology.