In this work we consider a monolithic approach that is based upon an arbitrary Lagrangian-Eulerian (ALE) finite element method. This conformal meshing approach permits the deformation of a single mesh throughout the domain, evolving with the fluid-structure interface. The importance of using an appropriate pressure space approximation to accurately capture the discontinuous pressure at the fluid-structure interface will be demonstrated and described. The stable Taylor-Hood P2/P1 element pair, widely used for purely fluid cases, will be compared against the lower order but discontinuous pressure pairing, P2/P0. These are both compared to the P2/(P1+P0) pair, which enriches the Taylor-Hood discretisation with a constant pressure on each element. Further, efficient algorithms employing block-preconditioned Krylov-subspace iterative methods are well known for the fluid flow problem. We consider the extension of these techniques to the discrete FSI problem, where the monolithic linear system contains contributions from the solid model as well as the fluid. Numerical examples will be presented examining the performance and stability of different solving strategies of these computational algorithms applied to the discretised FSI, both in two- and three-dimensions. Further results will examine the performance of efficient iterative solvers, applying different block preconditioning strategies, specifically when applied to the fluid-structure interaction ALE framework.