For NFL, here is what I do on the full Sunday-Sunday/Monday night slate, with the caveat that most weekends I am a break-even player, but have been profitable over the course of a full season by hitting a couple big scores each NFL year with some modestly decent scores mixed in.
In sum, you should identify a group of players you want exposure to, and since you know Solver, use that group of players to make multiple lineups. Specifically:
1) Start with QB. Personally, I try to determine which QBs have a reasonable chance of being the highest scoring QB in a given slate, and include all of them in my pool (no more than 10 QBs). I then pair each QB with at least 1 of his WRs. I have at minimum 1 lineup with each of the QB/WR stacks I have identified as potentially high scoring. While making 10+ lineups seems like a lot, IMO it's better to make 20 $1 lineups than one $20 lineup in the millionaire maker if you want to build a bankroll.
2). Paying up for RB is almost never worth it. Identify the high-floor/high-ceiling value plays at RB each week, which will probably be chalky, but who cares. This past week for me it was Sims III and CMike.
3) For WRs, 1 will be paired with your QB. The other 2 are up to you, and this for me personally is where the most variance in my lineups comes from. Don't worry about ownership, and don't worry about floor -- identify WRs who could potentially be the highest scoring WR that week, as well as ones who could exceed their value by 5x. Think long and hard about whether or not that player truly has the potential to either be the top scoring WR or to hit 5x value -- look at the matchup, the vegas over/unders and spreads to get an idea of game script, the players' recent performance, trends in their targets/red zone targets, any news about gameplan that a coach might implement for a given opponent that are different than the norm, etc.
3) TE -- use more or less the same process as for your 2 un-stacked WRs. These days the top-scoring TE almost never comes from the same group of elite players, so go deep in the range you will consider. You need to think outside the box of Olsen/Gronk/Reed, and dig deep to find value plays here that still have a high ceiling (Zach Miller this past weekend was a perfect example -- cheap now b/c of recent performance, but had some big games last season). Use targets, red zone targets, injuries/depth chart developments to help with this.
4) DST --- the chalk play here is almost never the right one, so throw all the D/STs that you think could have a big day into your player pool. Depending on the rest of your player pool, you may need to go cheap here, but that's not always a bad thing.
5) K--don't fret about this, but generally pick a kicker who is on a team that is favored at least. Avoid kickers playing in bad weather or rain. Give a boost to those playing in a dome. If you really want ot dig deep, though this is highly unlikely to win you any GPP, you could look into red zone offense vs red zone defense in a given matchup to estimate how many field goals a team might attempt.
HTH and feel free to PM me if you have any more questions.