Check the top 5 results for that keyword. Analyze the content, backlinks, domain authority, social sharing and so on. Then create content with more better and updated information. Also do work on backlinks, promotion and other things. Promotion is very important because it doesn't matter how great your content is, without promotion no one cares.
Relevant quality backlinks and Proper On-Page Optimization will helps to increase your possiblity of Ranking. Also, analyze your competitors site to find some more new method to overcome your competition.
The amount of on-page optimization you do when targeting a specific keyword is only half the battle.
That articles with more words, better URL ratings, and more backlinks will have an easier time ranking well for multiple keywords. The big takeaway though should be what the multiple keywords have in common.