Electrical control of magnetic skyrmions, which are topologically stable particle-like spin textures, holds promises for emerging memory and computing applications such as racetrack memory, logic devices and neuromorphic computing. Among the main advantages of magnetic skyrmions are their stability and non-volatility at room temperature, the low energy requirement for the motion, their sub-microscopic size and particle-like behavior. On the other hand, to perform neuromorphic computing with electrical devices, an architecture integrating both artificial neurons and synapses is required. The neurons operate a weighted sum and a non-linear activation function and the synapses store the synaptic weight with non-volatile tunability. To date, all existing proposal of electrical neuromorphic computing devices use artificial synapses and neurons which are physically separated. Therefore, macroscopic connections between synapses and neurons are required, which limits the miniaturization. In this study, we propose to use fully electrical control of skyrmions in compact devices made of magnetic multilayers to implement locally the main neuromorphic computing operations. The devices are based on the electrical control of a number of magnetic skyrmions injected into a detection zone as the weighted sum of the current inputs and the synaptic weights, and the non-linear electrical detection of their number. Using electrical current pulses with controlled current density and time duration, we show how it is possible to nucleate at room temperature a desired number of magnetic skyrmions and move them at a desired velocity within micron-wide tracks. We show that skyrmions moving within a Hall cross can be detected in real-time electrically by the Anomalous Hall effect, while these skyrmions can also be tracked by magneto-optic Kerr effect (MOKE) microscopy. Moreover, we also investigate how the tuning of the magnetic anisotropy and Dzyaloshinskii-Moriya interaction of the magnetic multilayer stack using electrical field manipulation with ionic liquid gating can be used to induce non-volatile modification of the nucleation and motion of the magnetic skyrmions in the devices. The long-term objective is to use all these basic knobs in order to perform neuromorphic functions using the assets of magnetic skyrmions