"What are you doing?" "The household ledger. Our finances are tight, so I've got to be strict about this kind of thing."
Literal
What am-doing? / Household-ledger [obj-を] [ね]. Our household [topic-は] tight-tight is-[because-から], this-kind-of place [from-から] properly-do [if-not-ないと].
家計簿 is the traditional Japanese household account book — many families keep meticulous records. カチカチ is a mimetic that can mean 'rock-hard, stiff, frozen'; here applied to finances it means 'very tight, no give.' ちゃんとする ('to do properly / get in order') is a versatile verb. The final ~ないと ('if I don't') is the truncated conditional with the consequence left implicit: '...I have to.' The elided を on 家計簿を along with the trailing ね gives it that spoken, unfinished quality.