I've not been to a match in London, so can't really comment on the ease of getting to see West Ham, Spurs, etc. I would suspect you can get in, though that will depend on who they are playing. The nearest non-London team is Portsmouth, and you should be able to get tickets to see them, but I've not travelled there by train from London. (You might find
this book interesting background reading, though, on both Pompey and visiting most of the Premier League grounds in general.) If you can't get tickets to a Premier League match (absolutely no-one in the UK refers to the 'EPL', and I suspect most won't even know what it stands for) Reading is in striking distance from London (easier than Portsmouth, I would say), and the chances of getting a ticket would be pretty high, and prices lower.