Naice parts of Brooklyn and Manhattan rents are on a par with each other atm. We paid 4K a month for a 2 bed rental and that was 2 years ago. Lots of rentals will include the cost of the hot water and heating if you live in a block - downside of that is you can't usually control when it comes on!
I find living costs are more expensive than the UK- there's a monopoly on gas and electric so you're stuck with one provider, and the same for cable / internet. I think our monthly cable bill with a high end high speed broadband package is ~$160 a month. Mobile phones the same irt sporadic choice (altho its much better than it used to be) and coverage can be piss poor even in parts of the city, due to the high rises.
Monthly subway pass is $121 - this will take you anywhere in the city, as its not zoned like London
renters insurance is about $30 a month
We budgeted $200 a month for gas and electric
I'd second want2's comments on the amount of tax to budget for, and the cost of food, especially if you're used to fresh veg, fruit and meat. Also I find booze and wine expensive - no more 3 bottles for a tenner like you used to get in the supermarket in the UK!
I will caution on the amount of money you'll need in a bank to secure a rental as its difficult due to lack of credit history to secure a rental with the usual month down + deposit. For recent immigrants with a lack of banking history its common to pay 6+ months rent upfront plus a month's deposit, plus the real estate agents fee, unless your work is acting as a guarantor for you. I had to find ~$11,000 upfront to get an apartment when I first moved over 