Mitochondrial membrane potential ($\Delta \Psi m$) in cells is a critical biological parameter that is related to diseases such as cancer through biological processes including metabolism, oxidative stress and apoptosis. Studying $\Delta \Psi m$ in normal and diseased states is limited by significant challenges associated with the current state-of-the-art methods for measurement, particularly fluorescent dyes, which are often toxic and photobleaching. The aim of this work was to develop electrical and chemical sensing methods to enable sensitive detection of changes in $\Delta \Psi m$ in single cells. Fluorescent Nitrogen Vacancy Centres (NVCs) in nanodiamond were identified as potential sensors of electric field that could be applied in living cells to directly measure $\Delta \Psi m$. Furthermore, Raman spectroscopy was identified as a label-free chemical sensing technique that could be used to reveal the location and chemical composition of the mitochondria in cells. To achieve the aim of detecting changes in $\Delta \Psi m$ using these two methods, complementary NVC and Raman measurements in live cells were needed. Initially, a specialised microscope was designed, built and validated to enable dual measurement of both NVC fluorescence and Raman spectroscopic signals simultaneously. Next, protocols and analysis techniques were developed for live cell Raman microscopy using a commercial reference instrument. An investigation of the impact of nanodiamonds as nanoparticles for biological sensing within live breast cancer cells, including surface modification by oxidation, was then conducted. By developing both techniques together, this project leverages the complementary advantages of long time course measurement, common laser excitation, the observation of nanodiamonds via Raman signal, with live single cell interrogation. The thesis concludes with an outlook on the future development needed to utilise NVCs and Raman spectroscopy in the measurement of cellular $\Delta \Psi m$. Achieving this goal in future will provide new biological understanding of $\Delta \Psi m$ in normal and disease states, allowing us to follow over time changes in $\Delta \Psi m$ in response to biological processes such as apoptosis in single cancer cells.