i think that TF is the hardest one to trick

Why in the world are you worried about increasing those things? In what way does that benefit you?

There are many thinks you can do but if you are looking for something automated that doesn't require time and work (or money) there isn't any.

First you need to focus on the content of your site, for example you can start writing resources on specific subjects on your niche trying to make them as informative as possible (check other pages/posts on the web and try to make yours more informative and with more details and add references to other relevant posts of authority sites on the web ) .

After that you will have to promote that content mostly via social media and even contact sites owners that you used as reference , you can just tell them about the great information they share and that you linked to their site (without asking anything in return). People really like when you share/link to their site and if you resource is really informative and adds value many will link to you in posts that they will publish in the future and are relevant to your resource.

Other things you can do is guest posting, blog commenting (not anything spammy and without any keywords for your links to your site) and other content marketing strategies!

That's not an easy job but on the long term you will see a big improvement on your site's metrics and rankings (you could see results within the first month).

I use Moz TrustRank and Majestic TrustFlow as main pre-qualifiers. After that, a look at most of the sites existing links to make sure they "mostly" look in good shape and not going anywhere soon. Most sites end up having good metrics in all areas anyways. But at this stage you can begin to see sites where page rank (PR), is no longer reflecting the pages true SEO value.

I find Moz a bit slow tbh. They have been a bit faster lately.

You raise it with backlinks just like with pagerank, find sources with high citation and trustflow and get links from there.

Obviously the guy is trying to increase those metrics so that he can sell links since its a new source of link metrics.

I like this thread.