The ASVAB score is more like a placement test. If you get a higher score you can get jobs which require a security clearance (ie: military intelligence), but if you have a DUI, I don't know how likely that would be. They have relaxed a lot of the enlistment rules since I was in (2001 - 2003) so I'm not sure. Some jobs also are more likely to come with an enlistment bonus option because they need more people to fill them and there aren't many people that want to do those jobs or that qualify with their ASVAB scores.
As for the enlistment bonus, I don't think it matters what your criminal record is if you've already been accepted. However, you wont get the bonus until after you complete all of your training, and if you leave the service before your enlistment contract date expires, you'll be required to pay back that bonus. They may also offer a GI bill contract, but keep in mind you will only have a certain number of years to use it and you'll be paying $100 out of your paycheck every month for the first year of service. So it's good if you actually know you'll use it - however it only pays for tuition... not books, parking, or other fees... and you're still required to pay up front and wait for reimbursement. I got the GI bill but ended up going with the VA's vocational rehabilitation program which pays for a certain amount of years toward the degree of my choice (as long as it is in a major that will provide me a job - I chose Biology) and it also pays for ALL of my college expenses without me having to worry about reimbursement. It might be different for disabled vets (I am one), but my GI bill just went to waste.
