You say High Competition...this is over broad. The only competitors in the race are the ones on the first page. Who are your competitors? Research the on page, power of their backlinks i.e. Dmoz & Yahoo. Are they authority sites? Are they EMDs also?
General tips: Keep your content very relevant, only rank 3 keywords for the first page, the EMD will get you most of the way there. Keep fresh content.
Develop a back linking plan to out rank the weakest one on the first page. Build from their one position at a time.
High competition or low you do the same for all sites. Just takes less effort for low comp sites.