Renters who pay rent are often curious about how their rent payments show up on their credit report. Your rent can be reported to the credit bureaus to help improve your credit report and to also show a potential creditor that you pay your rent on-time. You cannot do this yourself because it would have to be the creditor or in this case the landlord who collects the payments. This can be a fairly easy process for your landlord but it will cost them a lot of money.
Rent payments are what they call "alternative credit data" in the credit world. Things like your monthly utility bills would also fall in this category.
These items are not automatically reported to credit agencies like the mortgage or a car loan would. The reason being is that the cost associated with proper reporting can be pretty high. Your landlord would have to subscribe to each of the three credit rating agencies by paying hefty fees and being compliant with their monthly reporting procedures.
This is often too costly for most landlords. So, it is highly unlikely that you can have this done.