Dermatological disorders constitute a significant global health burden, yet the accurate and timely diagnosis remains a challenge due to the subjective nature of visual inspections and the lack of specialists. Although deep learning (DL) has shown promise in automated classification, the black box nature of these models and their high computational costs often hinder clinical adoption. This study proposes a lightweight, interpretable Custom Convolutional Neural Network (CNN) for the multi-class classification of five distinct skin conditions: Acne, Vitiligo, Hyperpigmentation, Nail Psoriasis, and Stevens-Johnson syndrome/toxic epidermal necrolysis (SJS-TEN). Using a data set of 9,548 images, we implemented a comprehensive pipeline featuring a two-stage augmentation and balance strategy to mitigate class imbalance. The new proposed SkinNet model was benchmarked against four established architectures: ZooCNN, CNN, MobileNetV2, and DenseNet121. Experimental results demonstrate that SkinNet achieved superior performance with an overall accuracy of 86\% and a macro-averaged F1-score of 0.85, outperforming transfer learning baselines. To address the critical need for transparency in medical AI, the model’s decision-making process was validated using four Explainable AI (XAI) techniques: Local Interpretable Model-agnostic Explanations (LIME), Gradient-weighted Class Activation Mapping (Grad-CAM), Integrated Gradients, and Diverse Counterfactual Explanations (DiCE). These visualizations confirmed that the model consistently focuses on relevant pathological features such as inflammatory papules, depigmented patches, and onycholysis rather than background noise. This research highlights the efficacy of task-specific, lightweight architectures to provide accurate, efficient, and reliable diagnostic support for dermatology.