My kids & I live with my BF & I pay for half of our living costs. It's a home, not a rental. Can I still claim head of household and claim that I paid rent for the year?

Am I Head of Household?

https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household

https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household

If you qualify as Head of Household, when you enter your marital status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.

Yes, if you provide more than one half of the living expenses for the household (you and your children in this case for tax purposes) you can claim Head of Household.  You do not have to claim it as "rent" necessarily, just try to keep good records of amounts you contribute (for example, if you give him cash and he makes the mortgage payments be mindful that you have a record of an atm withdrawal or venmo.  Or in other situations, you pay more for groceries or pay utility bills and not necessary pass that cash- you just want to be sure you can show you contributed).