There is compelling observational evidence for the existence of dark matter in the Universe, including our own Galaxy, which could possibly consist of weakly interacting massive particles (WIMPs) not contained in the standard model (SM) of particle physics. WIMPs may get gravitationally trapped inside heavy celestial bodies of ordinary matter. The Sun is a nearby candidate for such a capture process which is driven by the scattering of WIMPs on its nuclei. Forming an over-density at the Sun's core the WIMPs would self-annihilate yielding energetic neutrinos, which leave the Sun and can be detected in experiments on Earth. The cubic-kilometer sized IceCube neutrino observatory, constructed in the clear glacial ice at the Amundsen-Scott South Pole Station in Antarctica offers an excellent opportunity to search for this striking signal. This thesis is dedicated to the search for these solar dark matter signatures in muon neutrinos from the direction of the Sun. Newly developed techniques based on hit clustering and hit-based vetos allow more accurate reconstruction and identification of events in the detector and thereby a stronger rejection of background. These techniques are also applicable to other IceCube analyses and event filters. In addition, new approaches to the analysis without seasonal cuts lead to improvements in sensitivity especially in the low-energy regime (<=100 GeV), the target of the more densely instrumented DeepCore sub-array. This first analysis of 369 days of data recorded with the completed detector array of 86 strings revealed no significant excess above the expected background of atmospheric neutrinos. This allows us to set strong limits on the annihilation rate of WIMPs in the Sun for the models probed in this analysis. The IceCube limits for the spin-independent WIMP-proton scattering cross-section are the most stringent ones for WIMP masses above 100 GeV. / IceCube