mmWave radars have recently gathered significant attention as a means to track human movement within indoor environments. Widely adopted Kalman filter tracking methods experience performance degradation when the underlying movement is highly non-linear or presents long-term temporal dependencies. As a solution, in this article we design a convolutional-recurrent Neural Network (NN) that learns to accurately estimate the position and the velocity of the monitored subjects from high dimensional radar data. The NN is trained as a probabilistic model, utilizing a Gaussian negative log-likelihood loss function, obtaining explicit uncertainty estimates at its output, in the form of time-varying error covariance matrices. A thorough experimental assessment is conducted using a 77 GHz FMCW radar. The proposed architecture, besides allowing one to gauge the uncertainty in the tracking process, also leads to greatly improved performance against the best approaches from the literature, i.e., Kalman filtering, lowering the average error against the ground truth from 32.8 to 7.59 cm and from 56.8 to 14 cm/s in terms of position and velocity tracking, respectively.