The elastic proton-proton scattering data analyzed in this thesis was taken at the LHC with the ALFA detector at sqrt(s) = 8 TeV with a beta* collision optics which gave access to the Coulomb-Nuclear-Interference region. An improved procedure was used to fit the observed number of elastic events as a function of the four-momentum transfer with a simulation based on theoretical expectations including experimental effects. The data can not exclude any of the investigated parametrizations for the modulus of the nuclear amplitude. The fit results for the total proton-proton cross section, sigma_tot, and the ratio of the real to imaginary forward elastic scattering amplitude, rho, are: sigma_tot = 99.1 +/- 0.3 (stat.) +/- 1.6 (syst.) mb rho = 0.166 +/- 0.017 (stat.) +/- 0.019 (syst.) which are in agreement with previous measurements. As part of the analysis, the detector distance to the beam was determined with an improved method. A new background estimation procedure was developed and also implemented in the determination of the elastic track reconstruction efficiency. The experience gained in this thesis was used to optimize the experimental settings for a new data taking at sqrt(s) = 13 TeV with beta* = 2.5 km which will be used to determine the absolute luminosity for ATLAS.