Cheapest and easiest is to fly from Heathrow to JFK. Depending on what your plans are. You could also try Newark, which is better in my opinion, than JFK. I think the last time JFK was updated was when JFK was born. It's also out of the way.
I wouldn't recommend convincing yourself about getting a boat over, it's definitely something that's not a regularly cheap option, if at all. Container ships are expensive and take weeks. You can be in NYC in 8 hours if you fly.
Lastly, speaking as a Brit who has spent quite a considerable amount of time in the US in the past, don't get your hopes up. I don't want to kill the dream, but what you see on TV is not what is real.
Imagine all of those American tourists arriving in London thinking it was going to be like Downton Abbey! I kid. But still...
You are in the UK. It's quite an amazing collection of countries to experience. Get yourself a tarp and stick your thumb out. It's not as hard as you may think and you will have a great time.