I’ve read that CuSO4 solution reacts with peptide bonds that connect amino acids to create a violet colour, but the instructions always tell me to add NaOH solution to the protein solution before I add CuSO4. How is alkalising the protein solution before adding CuSO4 solution an aid to this process?
This is probably to prevent precipitation of copper hydroxide (see Itzhaki & Gill, 1964 - they suggest adding dilute copper sulfate slowly to the NaOH solution to avoid this). If you have the protein already alkalized and ready to react you'll get the color reaction before precipitate forms.
Commercial premixed solutions, like @canadianer mentioned in a comment, have tartrate present to prevent this (Geiger & Bessman, 1972 mention this).
It doesn't seem like it's strictly necessary to add NaOH first, but that's probably the most reliable way to do the assay without using an additional stabilizing agent.
Itzhaki, R. F., & Gill, D. M. (1964). A micro-biuret method for estimating proteins. Analytical biochemistry, 9(4), 401-410.
Geiger, P. J., & Bessman, S. P. (1972). Protein determination by Lowry's method in the presence of sulfhydryl reagents. Analytical biochemistry, 49(2), 467-473.
You mix those two chemicals to get a Cu(OH) 2 colloid, if you mixed them earlier it would settle down and not form a colorful complex (as efficiently). Base is added first because (i presume) it denaturates the protein which partly unfolds it, allowing copper ions to form complexes along a greater surface, giving clearer results.