Support Vector Machines (SVM) are widely used techniques in the field of classification problems because of their ability to effectively deal with datasets that have complex non-linear structures and a high dimensionality. The compute-intensive training algorithm associated with SVM makes it challenging to keep an up-to-date model that accurately reflects the characteristics of newly arriving data points in real-time systems. This paper proposes a novel training algorithm for incremental learning from large datasets, based on a variant of Sequential Minimal Optimization (SMO). High-Level Synthesis (HLS) was used for implementing the Field Programmable Gate Array (FPGA) based Intellectual Property (IP) Core, which includes the computationally intensive kernel computation portion of the training algorithm. In addition to the kernel computation, the inference phase of the SVM classifier is built into the IP core, and its use can be switched on the fly. The computational latency and memory bandwidth of an IP core are optimized using loop pipelining and DMA burst data transfer. With the help of hardware/software co-design, the IP core is integrated into the design of a flexible and re-usable System on Chip (SoC) called PYNQ Overlay. The experiments show that the overlay outperforms the embedded processor, multiple hardware SVM classifiers, and hardware accelerated Convolutional Neural Networks (CNN) in terms of real-time efficiency. The Overlay makes much less use of the resources available on the chip in comparison to the majority of the CNN accelerators. The overlay achieves an average classification accuracy that is only 1% lower than that of an ARM Cortex-A9 processor, according to experimental results on six datasets. Furthermore, it can increase training speed by an average of 31.82x and inference speed by an average of 31.74x. In addition, the proposed Overlay design achieves a 2.3x improvement in average training speed, as measured in Mega bits per second, compared to existing SVM training implementations, along with incremental learning and multi-class classification support.