He pays rent which is £90 a week, water which is £10 a week, sky which is £22 a week and TV licence which is £6 a week
So he's allegedly paying out around £500 a month?
Yet that excludes CT, utilities and food? 
I think you need to see the actual bills and rent statements for yourself OP to be on the sure side because some of those numbers don't sound right to me.
I pay roughly £600 pm for essentials - rent, bills (incl my £5 mobile) and food....and i live on my own with 2 cats!
You're a family of 4 so your basics (excl Sky) will be higher.
The only basic he pays is the rent whilst you pay the rest - does that sound fair to you?
It doesn't to me.
I suggest you each pay 50% of the total bill for essentials - rent, utilities, CT and food.
You chose to move in with a partner who has kids, he's lost his 25% CT discount and costs have increased - you can't expect to just pay 1/4 of essential bills/only for yourself - that takes the piss.
Luxuries like Sky need to be reconsidered and you can find a cheaper package.
I don't think he's being honest about the actual costs he's paying and so your fears about being taken advantage of may well be correct....
From my experience -
Rent - is usually calculated per calendar month so you pay a fixed amount each month. £350 per month is what i pay for a one bedroom place.
So £360 rent for a family home doesn't sound right....even if it was a slum landlord....
Water rates - mine have always been included in the rent, now it's included in my CT so i can't really comment on this.
Utilities - When i was on a prepayment meter it was £10 per week each for gas and electric for basic use. I lived on my own and was out at work all day.
A family of 4 will use a lot more....
Council tax - he will have lost his 25% single person rebate when you moved in. How much is the total monthly CT bill? I'm one band up from the cheapest and mine is around £100 a month with my discount....
Tv licence - don't own a TV.
Food - i'm single and live alone so it's cheap.
How much do you pay per week - incl when the dc are there?
Sky - £80 odd a month is INSANE!
I pay £20 a month for unlimited broadband and watch all my tv/movies via my internet. I use a site similar to Netflix but it's free so i don't even have to pay a subscription.
All the latest stuff is uploaded on there.
I also don't pay a TV licence because i only watch catch-ups when i get in from work.
Living on my own, i pay around £600 pm to cover my basics.
If a bf moved in, i'd lose my CT discount but it would work out cheaper for me as i'd be paying 50% instead of my current 75%.
Your bf earns £1500 a month?
Pays £500 a month so has £1k leftover....yet defaults on his minimum debt repayments and can't afford household items - so where's the other £1k going?
All this - and bearing in mind his comment regarding the crisps - i think he's using you to subsidise his lifestyle and expenses.