Let's start with the vitamin C itself. The pure form of vitamin C or L-Ascorbic Acid is extremely hard to work with. You can't just take a crushed up vitamin C tablet and add it to say your moisturizer. If you dissolve it in water, it will just crystalize when you apply the mixture to your face. In this form, it won't penetrate your skin and it won't provide any antioxidants the regular serum does. So you have to choose a carrier which will penetrate the skin such as alcohol, glycerin, or propylene glycol. Unfortunately, alcohol is drying and the other two when mixed with vitamin C do not penetrate the lipid layer of your skin very well.
Next you have the issue of having the serum at the proper pH. Since L-Ascorbic Acid is acidic but the skin runs between 5.5 and 6.5 which is closer to neutral. When you mix the L-Ascorbic Acid with water to form a 5 percent concentration, it ends up with a pH of between 2.2 and 2.5 which can cause problems with the skin but most commercial vitamin C serums run between 2.8 and 3.4 because it works but won't cause any irritation. If you make your own serum, you need to check the pH to make sure it is not too acidic.
Furthermore, if L-Ascorbic Acid is extremely unstable and will oxidise so it is not as effective when you expose it to light, heat, or air. When the vitamin C serum goes bad, it turns brown. So if you want to make your own, you have to do something to stabilize it. This can be done by adding antioxidants such as vitamin E and ferulic acid which help it last longer or you have to cover the solution with a protective coating to keep it from being exposed to light, heat, or air. Both solutions are difficult to do at home so you'd have to make a very small batch, store it in an opaque container, and refrigerate it but it only lasts 3 to 4 days.
Finally, most of us want more than just L-Ascorbic Acid, water, and glycerine. We want it full of antioxidants because antioxidants work together to be more effective and it increases their ability to protect you from the sun. Unfortunately, when you add extra ingredients, they can change the pH, make the mixture unstable, or clump. The other thing is that when you make vitamin C serum at home, you can't tell how effective it is because you do not have the lab facilities needed to test the recipe.
If you really want to make your own, you can find recipes but not all of them work. You really are better off buying the serum because they've been properly made. Let me know what you think, I'd love to hear. Have a great day.