First off, can the minor league players be kept, or is it just 4 roster spots during the year of non-major leaguers? If the minor leaguers are kept, then dive right in, and try to get the best prospects. For example, if you didn't think your team would win, keeping an Alex Gordon, even though you only keep 4 would be worth it.
It is tough to build with just 4 players. I would reccomend trying to take the Best player available in every round this year. Look towards young guys with upside, don't worry about backups, or having enough at positions, your goal is to get the best 4 at the end of the year.
You can go after closers if you want, but only think of them as trade bait, because you won't keep a closer.
Throughout the season know which players you think are your four best and work hard to improve those 4. So be willing to overpay, in order to upgrade your keepers, because the rest get thrown back anyway.
Also, target injured players on teams that are competing for the title this year. Last year (for example), Derrek Lee would have been a good target. Look for someone like that, who is an established star, but not helping this year.
Watch the waiver wire, you probably won't find someone who you will keep at the end of the year, but knows? Also, if a player gets hot you can use him to upgrade one of your keepers.