First, I mean no disrespect in any way, shape, or form. Second, you are smart enough to reach out and ask about this which is more than most people would do. Third, if you haven't dealt with enough storm damage to understand how to get that tree down safely, then I'd let someone else do it for you. The tree isn't huge, but its big enough to easily crush you. Obviously you know how to run fence, so plan on doing more of that instead. Oh, and short of having a piece of equipment that can pick the bottom end up and push it sideways into the brush, that fence IS going to get crushed so plan on fixing/replacing it regardless.
If you feel that you absolutely MUST get this down, start with the top. Clear all the brush around it so you have room to work. Get ALL the top off first. With the top on the ground, it will be more difficult to anticipate what wood is in tension and what's in compression, so take small bites. What I mean by that is start at the very top and take off the little stuff that's easy to get to. Keep things cleared as you go so that you're not tripping over things. Once you have all the obvious stuff take off, start working your way back towards the trunk.
I would recommend AGAINST cutting any of the first 3 supporting branches. Its not that cutting them is wrong. Its that you don't seem to have enough experience to know when its safe to cut them and how, which is fine. Cutting from the top back means that if the tree does roll to one side or the other, you're out past it where it can't hurt you. If/when you get to a point where the tree could hurt you if it rolled, stop. Figure out which way it's most likely going to want to roll. Figure out a way to prevent it from rolling the other direction, and then cut from the safer side. Pole saws can be handy for stuff like this.
