Okay so, my main competitor is 1st and 2nd. Then there's one site that takes 3rd and 4th. And my site is 5th and 6th. So I'm the third site on the 5th result for my keyword. I've been in this position for a couple of weeks. How do I break the barrier and surpass my two competitors? I've tried everything, I just can't get any higher.
Unfortunately there is not a silver bullet. It takes time for the search engines to accept you, and good quality backlinks to convince them. Or you may never break the barrier for your keyword. Sometimes you need to drill down a little and try to be #1 for a phrase containing your keyword.
What's your site's url and the keyword you are trying for with these? We can't really give you advice unless we know those because every situation is different.
If you give us an idea (even via PM) about the keyword, I'm happy to help. Unfortunately, the answer could be that it's impossible, but I can't say without seeing the sites involved.
If your competitor with the same keyword is still above you even if you've tried doing all things that you can think of then it goes to show that you still haven't done things that they've already did.
Can you post your URL? Have you tried using Market Samuri? its a great tool for analysing your competition and might show you where they are doing better than you.