I've setup Octopress and want to use totally custom styling basically throwing out most of the standard styling that applies to the rest of the site.
I do want to keep reference to all responsive goodness though that is baked in.
What's the best way to setup my custom landing page?
You mean like this?
^- My Octopress site.
Blog is at: http://eduncan911.com/blog
Since Octopress uses Jekyll under the skin, it would be better to think about how to accomplish this in terms of Jekyll - and there are many ways to do this.
OT: Personally, I find Octopress' theme layout, includes and customs far too complex and too granular. They did this to make it highly customable; but in my view, it tightly couples every template to another template.
Now with that said, there are multiple ways to achieve this. I did answer a very similar question here:
Creating an octopress theme from a wordpress theme
It's Jekyll: therefore, just go create the page however you want it. Replace the /source/index.html with whatever design, html and css you want.
If you don't want the blog roll, or want to move the "blog to another directory", just move the current /source/index.html to, say, /source/blog/index.html. That's it.
As I noted in the answer above, the only import thing is to keep the YAML frontmatter and specify the "layout" as you want to use. For example, I have a layout called "homepage" that is far different than any other layout. My /source/index.html uses layout: homepage.
But even then, you don't even have to use the YAML - create your own raw html file as you see fit. It will be used when you rake generate. This is what I personally did at first. Then, I slowly split up the homepage into the /source/_layouts/homepage.html and just went from there. I did not follow Octopress' entire theme at all - just enough to use posts and pages.