Well, there's a reason that this trophy is the rarest one. Flappy Goat is a mini-game that is accessed in one of two places.
On the first level you can find it in the Coffee Stain HQ (a large house in the corner of the map near some other houses) upstairs on the hi-def TV. Or you can find it on the second level on an arcade machine under the ferris wheel (the arcade machine on the left of the three).
You start the game by walking up to it and then pressing triangle.
The game itself is a clone of Flappy Bird where you press one button to jump in mid-air and have to get your goat to pass safely through gaps. The problem is that it is initially balls-hard and also has very bad, and unfriendly, collision detection.
1. The TV version is in 16:9, the arcade version is in 4:3. Pick the ratio that suits you. I went with 4:3.
2. Find the rhythm that keeps the goat at a set height. You can get four jumps in between gaps so use that rhythm and slightly adjust for when you need to move up and down.
3. Turn the music/sound off. I muted my TV. You need to be able to keep that rhythm going.
4. Try to get a good view of the screen. I pulled a bench over to the arcade machine and stood the goat on that to get a closer view of the screen. I actually used an angled view where the left side of the screen was closer to me. I guess it helped.
5. It helps to get a little distracted. My best runs all involved me thinking about other things before my concentration snapped back and I ended up crashing on 8 or 9.
6. You can go from a good run to a series of rubbish ones easily. Do not worry about it. You'll get your mojo back as quick as it went.
7. Ignore people who say 'this was easy, I got it on like my third go.' These people are pricks. Every trick trophy attracts them.
8. If you meet anyone who works for Coffee Stain HQ, they've earned a warm palm across the face.
9. When you start, you'll likely struggle to get past 1 or 2. That is fine. This trophy isn't that rare which means that plenty of people have managed it. You will get better and when you find your rhythm, you will get this trophy and the sweet platinum. It's all good.
10. If you're going to downvote this solution, it is what it is. Don't hate the player, hate the game, kids.