Try and get in touch with an agent. The guys that I know who have played there, or even gotten offers have done so through their agents.
Difficulty on getting onto a team there will mostly depend on where you have played here. Some teams might be willing to give you a shot if you pay your way over during camp next season, but just calling up a team and saying you're a good hockey player isn't the most professional way to do it.
Ask around. Try former/current coaches and see if they know anybody. Like any other business, sometimes it comes down to who you know and who knows you.
What is your playing experience level? Here in the UK there are relatively strict work-permit restrictions.
This is from Ice Hockey UK (kinda governing body):
Permits for players or goaltenders can only be issued for teams competing in the Elite Ice Hockey League (EIHL). No work permits for players or goaltenders should be issued for any development leagues operating in the English, Scottish or Northern Ireland Associations
Criteria for players
First time players
Players coming to UK from USA or Canada
Players coming to UK from USA or Canada should be professional and have played in at least 75% of competitive games where they were available for selection from the following leagues, in the last complete season prior to the start date of the relevant current domestic season.
• East Coast Hockey League (ECHL)
• American Hockey League (AHL)
• National Hockey League (NHL)
• United Hockey League (UHL)
• Central Hockey League (CHL)
This means you would only be elegible to play (as a professional/semi-professional) in the Elite League which is a standard similar to ECHL. I am not sure what the deal is with the English Premier League (lower standard, "developmental league") if you play as a non-pro, but i don't think that is allowed. There are some other leagues that you can get in touch with in the English National Hockey League or the Northern Premier League where you may be able to play as an amatuer without a work permit.
Edit: That's all irrelevant if you have an EU passport aswell as an American one as some North Americans do because you wouldn't need a work permit and therefore could play in any league. If that is the case then many clubs have training camps you can register for and go to.