We present the results of detailed analysis of pointed X-ray observations by RXTE of the black hole X-ray binary (BHXRB) system V4641 Sgr (SAX J1819.3-2525) during its outburst of 2003 August. Soft X-ray (3-20 keV) flux variations by factors of 10 or more on timescales of minutes or shorter were seen. The rapid and strong variability of this source sets it apart from typical XRBs. In spite of large luminosity fluctuations, the spectral state of the source did not change significantly during the dwells, which suggests that the physical emission processes did not change much during the observations. The energy spectra during the dwells were dominated by a hard Comptonized power-law component, indicative of the canonical low/hard state observed in other BHXRBs. No soft thermal component was found in three out of the four RXTE pointings. However, spectral deconvolution of the observation with the largest average luminosity suggests an obscured, hot accretion disk. During one of the observations, we detected a short-term (~100 s) soft X-ray dropout, which is apparently due to variability in the observed column density. A strong Fe Kα fluorescent emission line near 6.5 keV was detected with large equivalent widths in the range 700-1000 eV. In the temporal domain, the Fourier power spectra were dominated by red noise below a few hertz. Poisson noise dominated at higher frequencies, and no high-frequency features were detected. The strong Comptonized spectra, broad iron emission line, absence of a disk component in the spectra, absence of any timing variability above a few hertz, and occasional large changes in the column density along the line of sight all support an enshrouded black hole with occasional outflow and a very dynamic environment.