Any chance without buying backlinks?

to appear more natural, it would be better to link to some good websites in your niche. The best way is to find a seo who has links from various themes. You give him links from your website(that will help you appear more natural) and in return he will give you some links from your niche websites that he has.

You are either going to have to spend a lot of a) time or b) money on the site.

Chances are the competitors who you have briefly mentioned are spewing out a lot of content and links on a regular basis. You must (at a minimum) meet, or exceed, whatever it is they are doing.

You probably could do it without buying links. It would take a long time to find the 'juicy' links available, make sure you are using keywords as the anchor text on the links your building and also linking relevant (inner) pages to relevant pages on other sites to pass the most link equity.