How do I increase my chances at making Sergeant First Class with derogatory paperwork in my file from a DUI?

You might try a defensive driving course and then write a letter to the board explaining how you've changed your ways and the things you've done to show that.

Boards are required to view derogatory information, no matter how old.
With your derogatory information you will probably never be eligible to hold a "position of trust" such as instructor or recruiter that will make your peers more competitive than you. So, you'll have to seek out other challenges. Being rated (and doing well) in a higher rated position is always an excellent way to show the board you are capable of doing the job. But, most likely that opportunity won't present itself unless your post is drastically understaffed on 35P4.
Next step is to take on additional leadership responsibilities or compete in leadership roles. NCO of the QTR/YR, SAMC, or other competitive boards always receive high marks from the board AAR's.
One of the best things you can do is find a need and fill it. My battalion was lacking in knowledge about the new BRS so we arranged briefings for family members. The post BSEP class was over full so our brigade made a deal to provide instructors if they certified us to teach our own people. Any way that you can be an asset to your unit will raise your value to them and stack your NCOER.
Then, of course there are the basics:
PT - consistently score 270 or above on all NCOERs
Civilian Ed - self explanatory
Military Ed - you should always be showing consistent self development
DA Photo - DON'T LOOK FAT

As for the other stuff, try to find things that benefit the Army or the community and make an impact at a higher level than your Brigade. Community volunteer work is good. Volunteer for taskings that require leadership. Maybe there is a crappy tasking to take a party ADVON to NTC, or be a detachment somewhere no one else wants to go. Volunteer to be the Color Guard NCOIC. By being the person who is the go-to guy you'll learn a lot and when career progressing opportunities come up, you are the one who will be recommended to fill those challenging roles.
If you're not Airborne yet, your Airborne colleagues have a higher rate of getting picked up, especially the ones with Special Operations time because it's a challenging assignment.

Performance, performance, performance.
NCOERs; NCOQ/Y; NCOES DHG/Leadership Award; SAMC.
Folks on here who have sat on boards can provide better advice with regard to how a board considered a DUI at that grade and length of time since.