I feel to obtain a greater level of cleanliness it may require magic. Is there a product anyone can recommend or a method of generating your own magically infused shampoo? Can magical shampoos exist in this world?
First question; what kind of cleanliness are you after? And what is your goal/purpose?
If you are just looking for physical cleanliness, then magic isn't needed. Just a balanced routine. Two-in-one shampoos are convenient, but their results are pretty lackluster. Use separate shampoo and conditioner. Wash and rinse your hair thoroughly, then apply the conditioner in a moderate amount (more is not better). Let the conditioner rest in your hair for at least five minutes, then rinse out. I start with the hair, put in conditioner, then leave it in while I wash my body and rinse at the end. Timing works out well. As a pro tip, also do not wash your hair every day. It wipes out the natural oils you produce, and then they over-produce which turns hair greasy. Rinse (no soap) and brush your hair every time you shower, brush once every day regardless of shower (smooths the oils) and wash about every three-or-so days. Your hair will thank you.
As for magical/energetic boosts, There are several ingredients you can apply to achieve what you are looking for. Some you can simply add into your usual shampoo, others would be better used separately as a rinse before or after you wash your hair. Different ingredients will have different effects, and different ingredients will have different methods of practical use.
For example Salt is a powerful grounding tool, and can be very effective on washing away negativity in many forms. It is also gritty so it doubles as a decent exfoliant. But that means be mindful of how much you use and how often. A pinch of fine salt added to the shampoo right before applying it to your hair can add some energetic (and scrubbing) Oomph. But, if you just dump it into your shampoo bottle and call it done, it will just dissolve and you'll lose the scrubby benefit. Though the energetic side would obviously remain.
For a power boost if you are preparing for intense magical work, you could add a pinch of dragon's blood Resin instead. Dragon's blood is about empowerment, uploftment, and authority. Three things important to bring to the proverbial table during certain kinds of magical workings. This could be incorporated into your preparation process before a working. ... For several reasons, this is not an 'every day' ingredient.
Herbs like sage, oregano, mint, vanilla, goldenrod, and many others can be used as well. Learn about the herbs and what their qualities are, then choose which herbs best apply to/support your goals. Though having plant bits in your hair all day can be uncomfortable so I recommend steeping any herbs you might want to use in water and using that as a rinse just before or just after the wash. Run a cup or so of the water tincture into your hair to wet it, wash and rinse as usual. Or if you like the smell of your creation put it into a small pump-spray bottle and mist your hair after your shower.