If you want your website to rank #1 for a given search term, there are a few steps you are going to need to take. 

Content

Without content around your chosen keywords, Google is not going to rank any site for anything related to the chosen keyword.  Obvious right?  So start by adding the content in the form of articles about the keywords.   

On Page Optimization

Make sure the content has the right amount of references to the keywords.  Too many and the site will be passed over, while too few does not work well either.  There are a number of free tools to check keyword density on a page by page basis.

Links

Assuming you have the keyword work done right, your site has some meaty content, your domain is not completely fresh, and there are plans in place to refresh and add to your content over time, the remaining variable is links. 

Your goal is to replicate the link profile of the sites that currently rank high for your most important keyword, then do that for the next most most important keyword.  These steps assume you have already done all the keyword research.

Step 1:  Go to Google and type in your first keyword.  Look at the first site that comes up.  Let's call it www.topsite.com

Step 2:  Open another tab and type into Google   link: www.topsite.com  

This results in a list of websites that link to topsite.com.  It is not a complete list, even of dofollow links, but it is a good start.  Make a list of these sites in a spreadsheet.  

Step 3:  Visit each of the linking in sites and find a way to get one link from that site to your site.  Ideally that link will be anchored by the main keyword.  In some cases it might be no more complicated than signing up and contributing an article to a site like infobarrel.

Can you guest post there?  Can you offer a link to their site?  How about offering blog post promoting their site?    Can you search the site for the keyword and suggest a couple places to insert a link to your site?

Keep track of your efforts on the spreadsheet - very important.

Step 4:  Repeat this process with the #2 ranked site for the top keyword and then the #3 ranked site and so on.  Work your way through the top 10 results or so, using your spreadsheet to filter out the sites you already contacted (some will show up linking to multiple sites in the SERPs)

Step 5:  Start the same process with keyword #2, rinse and repeat.

Step 6:  Can you get links off of any of the sites that show up on page 1 through 3 of the SERPs.  Even if they are not linking to other sites in the SERPs they obviously have authority for your keywords.

While this plan is clearly labor intensive, when completed your target site will be at the center of all sites with authority around your target keywords - a position that will be nearly impossible to change.