Skin cancer is the most commonly found kind of cancer with eight diagnostic classes, which makes its classification highly challenging. Recent years have witnessed the increased utilization of Computer-Aided Diagnosis (CAD) systems in several medical imaging processes, even in identifying skin cancer. Though multiple methods have been developed in the past to classify skin cancer, the high similarity of the skin cancer lesions with the surrounding skin makes classification a tedious process. A Deep Learning (DL) technique is introduced in this work for classifying skin cancer into eight classes based on skin images. This work presents two novel contributions to skin lesion segmentation and classification. Here, a technique for segmenting skin lesions is presented using DoubleU-Net, whose structure is optimized by the Sparrow Ladybug Beetle Optimization (SLBO). Further, a Fractional SLBO-Deep Batch-normalized eLU AlexNet (FSLBO-DbneAlexnet) is developed for classifying skin cancer. In addition to this, the competence of the FSLBO-DbneAlexnet in skin cancer classification is examined by considering accuracy, False Negative Rate (FNR), False Positive Rate (FPR), True Negative Rate (TNR), and True Positive Rate (TPR), and is found to produce superior values of 0.911, 0.080, 0.081, 0.919, and 0.920, respectively.