The hypoxic conditions within avascular solid tumours may trigger the secretion of chemical factors, which diffuse to the nearby vasculature and promote the formation of new vessels eventually joining the tumour. Mathematical models of this process, known as tumour angiogenesis, have mainly investigated the formation of the new capillary networks using reaction-diffusion equations. Since angiogenesis involves the growth dynamics of the endothelial cells sprouting, we propose in this work an alternative mechanistic approach, developing a surface growth model for studying capillary formation and network dynamics. The model takes into account the proliferation of endothelial cells on the pre-existing capillary surface, coupled with the bulk diffusion of the vascular endothelial growth factor (VEGF). The thermo-dynamical consistency is imposed by means of interfacial and bulk balance laws. Finite element simulations show that both the morphology and the dynamics of the sprouting vessels are controlled by the bulk diffusion of VEGF and the chemo-mechanical and geometric properties at the capillary interface. Similarly to dendritic growth processes, we suggest that the emergence of tree-like vessel structures during tumour angiogenesis may result from the free boundary instability driven by competition between chemical and mechanical phenomena occurring at different length-scales.