I have a mobile auto detailing business. I have an office at home with my computer where I do all my marketing/online advertising, calling customers, all that good stuff. The apartment that me and my wife live in is in her name but I make the rent payments. Can I claim a portion of those payments as a deductible business expense on my taxes ?
Assuming you are in the US, this should be standard home office rules.
You can claim only space you use EXCLUSIVELY for business -- if it's a corner of the living room, that means only the area occupied by your desk and chair, and even those only if you don't also use them for personal activity such as browsing Stack Exchange. Few people are really willing or able to carve out that space, but it can be done. (Easier if it's a separate room that you stay out of except when working, of course.)
Once you have established that space which is exclusively used as office, you can then pro-rate by the ratio of that space to the rest of the cost of maintaining the house or apartment... or you can use the Simplified Home Office rules, which don't require all that calculation and just say "deduct $5 per square foot of legitimate office space."
See https://www.irs.gov/businesses/small-businesses-self-employed/simplified-option-for-home-office-deduction and the documents referred to by that page.