How can I rank content better, and expand what it ranks for, without many backlinks?

I have a website that ranks decently for its main terms. For years I've been trying to get it to move up from #3 to #2, #2 to #1, etc.

But recently I've decided this is just a frustrating dream and I might not ever outrank the guy above me.

So I decided to look into my Clicky analytics and I see I have several pages that are getting a lot of traffic.

I would love to tweak these pages, improve their rank, wait and then tweak them some more and "Snowball" them to make them large pages that get a ton of traffic. I basically want to increase what they already rank for, but also expand the terms that they rank for.

I think a place I can start at is to use SEMRush to see what the pages rank for, and then try to weave the most common words into the H tags, etc.

I just fear over-optimization a bit.

Anyway, my question to you is this: Is there anyone who teaches this stuff? Some marketeer guru? I want to find a master on how to do this.
Also, is there any SOFTWARE that does this?

I'd love anyone's insight. Thanks a lot.

Also, as far as links I was thinking of doing some internal linking by using related pages as topical relevance. They'd point to the pages I want to rank. Then I'd also throwin a few strong links from outside of my site.

Also of course I'd add videos, authority links, length and other little tweaks. But I am just looking for someone who is smarter than me.. someone to follow online and learn some stuff from.