Fluid-structure interaction problems have a wide range of applications, but their efficient solution - in particular of the linear systems - remains challenging. In this work, we provide all details necessary for a monolithic arbitrary Lagrangian- Eulerian (ALE) implementation using the finite element library deal.II. Moreover, we show different ways of incorporating the continuity conditions on the interface on the discrete level. To actually solve the arising linear systems,we develop a preconditioner based on an approximate block-wise LU-factorization, splitting the coupled system of equations into its natural fluid, solid and mesh subproblems. Alongside these developments, we briefly review various linear solver techniques. Numerical results illustrate the robust convergence with respect to different material parameters and meshsize h, but with an acceptable dependence on the time-step size Δt. Furthermore,we observe that this iterative approach outperforms direct solvers even for a low number of degrees of freedoms and without parallelization.