There isn't a simple, mechanical way to translate a 5E character into 3E. The systems are too different. There's going to be a lot of adaptation and eyeballing involved.
In this case, the big issues you face are, first, finding 3E mechanics for dragonborn, and, second,
3E doesn't allow paladins to be any alignment except Lawful Good*. The most obvious solutions are to use the half-dragon template or dragonkin race (each of which, as
@payn notes, packs a hefty level adjustment) and the blackguard prestige class.
[Edit: Did not see the Races of the Dragon option, I thought there were no official dragonborn mechanics. Note however that the dragonborn of RotD are former humans transformed by magic, not a self-reproducing species as in 4E and 5E.]
Now, to be a blackguard, you need +6 base attack, 5 ranks in Hide, 2 ranks in Knowledge (Religion), and three feats: Cleave, Improved Sunder, and Power Attack. The simplest way to do this is with 7 levels of paladin, putting cross-class ranks in Hide. This lines up rather nicely for 20th level: 7 levels of paladin, 10 levels of blackguard, and +2 (dragonkin) or +3 (half-dragon) level adjustment. If you pick dragonkin, you can throw the leftover level into fighter or something.
However, if you try to replicate the 7th-level version, you're going to run into a number of problems. First and foremost, you have to be a minimum of 6th level before you can put any levels in blackguard, so if you want any blackguard abilities, you can't have a level adjustment as well. Furthermore, the requirement of 5 ranks in Hide cannot be achieved with six levels of paladin (you top out at 4), so you'd have to dip into some class that gets Hide as a class skill and also grants full BAB. Ranger is the obvious choice.
*You can be a paladin who used to be Lawful Good and no longer is; but none of your paladin abilities will work. You're basically a crappy fighter until you can qualify for blackguard.