The Wilkinson Microwave Anisotropy Probe (WMAP) satellite has provided high-resolution, high signal-to-noise ratio maps of the sky in five main frequency bands ranging from 23 to 94 GHz. These maps consist in noisy observations, a mixture of cosmic microwave background (CMB) anisotropies and of other astrophysical foreground emissions. We present a new foreground-cleaned CMB map, as well as a new estimation of the angular power spectrum of CMB temperature anisotropies, based on 7 years of observations of the sky by WMAP. The method used to extract the CMB signal is based on an implementation of minimum variance linear combination of WMAP channels and of external full-sky foreground maps on a frame of spherical wavelets called needlets. The use of spherical needlets makes possible localized filtering both in pixel space and harmonic space, so that the internal linear combination (ILC) weights are adjusted as a function of location on the sky and of angular scale. Our CMB power spectrum estimate is computed using cross-power spectra between CMB maps obtained from different individual years of observation. The CMB power spectrum is corrected for low-level biases originating from the ILC method and foreground residual emissions, by making use of realistic simulations of the whole analysis pipeline. Our error bars, compatible with those obtained by the WMAP collaboration, are obtained from the combination of two terms: the internal scatter of individual Cℓ in each ℓ bin, and a term originating from uncertainties in our correction for biases due to empirical correlations between CMB and foregrounds, as well as to residual foregrounds in the CMB maps. Our power spectrum is essentially compatible, within error bars, with the result obtained by the WMAP collaboration, although it is systematically lower at the lowest multipoles, more than expected considering that the two estimates are based on the same original data. Exhaustive investigations of the presence of a possible bias in our estimate fail to explain the difference. Comparison with several other analyses confirms the existence of differences in the large-scale CMB power, which are significant enough that until the origin of this discrepancy is understood, some caution is recommended in scientific work relying much on the exact value of the CMB power spectrum in the Sachs-Wolfe plateau.