We present the observational results of an L'- and M-band adaptive optics imaging survey of 54 nearby, Sun-like stars for extrasolar planets, carried out using the Clio camera on the MMT. We have concentrated more strongly than all other planet-imaging surveys to date on very nearby F, G, and K stars, prioritizing stellar proximity higher than youth. Ours is also the first survey to include extensive observations in the M band, which supplement the primary L' observations. Models predict much better planet/star flux ratios at the L' and M bands than at more commonly used shorter wavelengths (i.e., the H band). We have carried out extensive blind simulations with fake planets inserted into the raw data to verify our sensitivity, and to establish a definitive relationship between source significance in {sigma} and survey completeness. We find 97% confident-detection completeness for 10{sigma} sources, but only 46% for 7{sigma} sources-raising concerns about the standard procedure of assuming high completeness at 5{sigma}, and demonstrating that blind sensitivity tests to establish the significance-completeness relation are an important analysis step for all planet-imaging surveys. We discovered a previously unknown {approx}0.15 M{sub sun} stellar companion to the F9 star GJ 3876, at a projected separation of about 80 AU. Twelve additional candidate faint companions are detected around other stars. Of these, 11 are confirmed to be background stars and one is a previously known brown dwarf. We obtained sensitivity to planetary-mass objects around almost all of our target stars, with sensitivity to objects below 3 M{sub Jup} in the best cases. Constraints on planet populations based on this null result are presented in our Modeling Results paper.