Speckle noise reduction algorithms are extensively used in the field of ultrasound image analysis with the aim of improving image quality and diagnostic accuracy. However, significant speckle filtering induces blurring, and this requires the enhancement of features and fine details. We propose a novel framework for both multiplicative noise suppression and robust contrast enhancement and demonstrate its effectiveness using a wide range of clinical ultrasound scans. Our approach to noise suppression uses a novel algorithm based on a convolutional neural network that is first trained on synthetically modeled ultrasound images and then applied on real ultrasound videos. The feature improvement stage uses an improved contrast-limited adaptive histogram equalization (CLAHE) method for enhancing texture features, contrast, resolvable details, and image structures to which the human visual system is sensitive in ultrasound video frames. The proposed CLAHE algorithm also considers an automatic system for evaluating the grid size using entropy, and three different target distribution functions (uniform, Rayleigh, and exponential), and interpolation techniques (B-spline, cubic, and Lanczos-3). An extensive comparative study has been performed to find the most suitable distribution and interpolation techniques and also the optimal clip limit for ultrasound video feature enhancement after speckle suppression. Subjective assessments by four radiologists and experimental validation using three quality metrics clearly indicate that the proposed framework generates superior performance compared with other well-established methods. The processing pipeline reduces speckle effectively while preserving essential information and enhancing the overall visual quality and therefore could find immediate applications in real-time ultrasound video segmentation and classification algorithms.