There is a super cool addon out there that will tell you who does what damage in fights, I know that - I hear it referenced when we run trials - but I don't have it, or know what it is. I'll have to ask for you. Heck, I should probably use it, too! We had a PVP (teaching) class in the UESP PC-NA guild, where I got the ones I use now. As far as addons, the only one I really rely on at the moment is
Srendarr, which displays buffs and the duration of attacks. Useful to me because it lets me see how long my skills and buffs last for, so I can make sure to keep them active. In Battlegrounds I just go by
the end of game recap to see the damage I've done.
I am sorry to say that even with that little "Yippee!", I am a
super casual PVPer.

Mostly I am PVE - in fact, I don't even really change my armor or my skill bar, for the most part, to do a match, which is lazy, I know,
I know - but dang it, I love the Battlegrounds. I go into them to break up the questing and to get some laughs, and I don't take them seriously at all (which probably helps me in the long run - I can just concentrate on having fun and not being on the leaderboard, which isn't hard to get on, anyway). PVP is all luck of the draw, really, especially in Battlegrounds. If you get stuck with a group who doesn't stick together or know enough basic tactics, it's unlikely you'll do good.
As for getting damage up... For the most part, on this character I actually researched different builds and started by going off of that. It helped me learn which skills were useful and I think helped me really understand the class better. My main is a Templar, which is the class I first started the game with, but man. I notice a big difference between my characters that followed/borrowed heavily from different builds in terms of sets and skills and my characters from when I first started the game waaaaay back in the day; those latter characters can do PVE content, definitely, but they struggle more. I really thought dying often was pretty common back then. After learning how the mechanics worked and running stuff with groups of people who knew their stuff and could share some knowledge, my game has totally changed. Especially PVE content. I might die in trials and sometimes in veteran dungeons, but there is nothing that I struggle with, and it makes the game
much more enjoyable. I can't tell you how much my life improved after I learned to actually use my Radiant Oppression skill.
Then it's the food and armor. Once I got a real set of armor (even just a basic crafted set) and buffed it up with enchantments, and then learned which food/drink combinations best worked with my skills, it all fell together. Since damage scales off your total magic/stamina, getting that up as high as possible helps. All my attribute points are in magicka, so having the buff on that and also my health to get it over 17k so I'm not
as squishy, really made a big difference. And one day I'll have farmed even better sets, so then it will be even better. But like I said, I'm pretty casual and play for fun, so as long as I can pull my weight in dungeons/trials, everything else is just to amuse me. You won't see any of my characters as the Emperor of Cyrodiil, that's for sure. I don't have that kind of time to endlessly grind Alliance Points.
The rest has been pure trial and error. I got some really good tips from that PVP class, especially from another player with the same class, as well as some armor (impenetrable trait is key for PVP). I like the Templar class because you can pack pretty high damage and also give yourself some super quick heals if you need them. I've won a lot of matches by putting down an AOE attack or two and spamming heals when I'm stuck, and I've lived to tell the tale more than once...try though they do to really get me, especially after I've played a few matches with them, bwahaha.
That's a super long post to say that I don't really have any secrets of the trade! I still spend most of my time doing solo PVE stuff or decorating my houses...because Homestead is the real end game. Haha.