What is the best method to push this site?

What is the best method to push this site in google: homepage-check.com
I tried several strategies but they don`t work very well.
Has someone an idea?

You should make Facebook ads with target - SEO, on-page optimization etc. It's not bad site btw.

I already tried Facebook ads. But didn`t worked well.

Fiverr - get a pretty video spokesperson to do a 45 second ad (for perhaps 3 gigs?)

AS for Facebook - I would go that way to. But only with video and only using Facebook's ad manager. NEVER with the boost button.
Use scrapebox to grab a custom audience of businesses likely to have websites (using keywords or the Yell scraper) then the email grabber to grab emails from those sites, then filter those emails for too many generic business emails (webmaster, info, sales, enquiries etc) put what's left in as a Facebook audience. Wait 20 minutes. If this audience is over 10,000 go with that with a simple headline "Is your website letting your business down" or something similar. If it's less than 10,000 use the look-a-like audience function and get it over 10,000 - but keep it quite tight. If you can get 100,000 all the better. All English speaking countries if that's your reach and website language

in Facebook PAY FOR CLICKS TO SITE ONLY and set the pay level LOW (8 cents is golden for me - you'll end up paying about 6 cents). You won't get millions of hits, limit spend to $5 or $10 a day until you have a feel for traffic and conversion. Scale up as/if your conversion level warrants it.

Thank you very much for this tutorial. I will try it out!

Why the craze about Facebook ads?

I have two questions for you:

1. Who is your site's target audience?

2. Where does your target audience "hang-out" online?

If you can answer these two questions, I think you can find where to promote your website.

BTW: Great idea for a website, good luck!

What else have you done beside the FB ads?

i optimized the site for the keyword "homepage check" but it doesn't rank for this keyword!?

There are 320 monthly searches (on a global level) with a 0.48 competition.

I suggest you try to rank for other kws

The OP should ask what kind of traffic is most wanted first, or most---SEO or search engine traffic, or direct traffic from related sites, other referrals, and from general branding? What is the site best geared for? Pursue the promotion plan based on those answers.