Dragonborn of Bahamut
The traditional (cheesy) approach is to use the dragonborn template from Races of the Dragon. Dragonborn is an LA +0 acquired template, not inherited, and can apply to warforged. It replaces most racial traits with those of dragonborn. Dragonborn are Humanoid (dragonblood).
The fact that dragonborn is an acquired template is what’s important here: a creature becomes dragonborn by undergoing the Rite of Rebirth, described in Races of the Dragon, a special ritual in which the creature dedicates him- or herself to Bahamut and is “reborn” as a dragonborn. Thus, no draconic ancestry is expected or desired for becoming a dragonborn. As a living creature, albeit odd ones, warforged would appear to be acceptable for this (and it definitely works RAW).
This does, however, touch on one of the great “mysteries” of Eberron, things the developers explicitly did not answer, leaving it for the DM to decide for him- or herself, and for the players to discover over the course of play. Specifically, the question of whether or not warforged have souls: Eberron Campaign Setting will not answer that question for you. Since becoming dragonborn is a very religious, spiritual exercise, lacking a soul would arguably prevent warforged from completing it. So if, in your DM’s game, warforged lack souls, the dragonborn template may be unavailable. (Even if they do have souls, there could still be an incompatibility, of course.)
Personally, I tend to rule that warforged do have souls, but if I were already planning a campaign where they do not, I would probably not tell the warforged, or tell him he can’t attempt the Rite of Rebirth: I would probably have the Rite fail, and that be an interesting dilemma for the warforged in question (was he not good enough? or can warforged simply not do it?).
It should also be mentioned that there is a balance concern with dragonborn warforged; I labeled it cheesy for a reason. It’s cheesy because one of the things dragonborn doesn’t replace is subtypes; this means a dragonborn elf is Humanoid (dragonblood, elf), and a dragonborn warforged is Humanoid (dragonblood, living construct). The living construct subtype is the source of almost-all of a warforged’s benefits, making the dragonborn warforged race extremely strong.
It’s not necessarily the best race always and forever, so it’s not necessarily “broken” or “overpowered,” but it is definitely and distinctly superior to 99% of races; dragonborn and warforged were already some of the better options, and the combination puts in the company of things like dragonwrought kobolds, lesser aasimar, and strongheart halflings.
If you are interested in becoming dragonborn because the concept of a dragonborn warforged sounds cool to you, and the way it hints at one of the great mysteries of Eberron interests you, your DM may wish to tone down what living construct does for a dragonborn warforged (eliminating it entirely, on the other hand, basically makes you no longer warforged at all, and defeats the purpose of the exercise).
Aside from dragonborn, the cheapest option is the Dragontouched feat from Dragon Magic. Its only requirement is 11 Charisma, and grants the dragonblood subtype along with a smattering of other tiny bonuses. It explicitly can reflect a spiritual connection to dragons, rather than an issue of heredity.
Most other options are hereditary, and, well, bad—they involve level adjustment, and that’s just categorically not worth it. One exception, ish, springs to mind though: dragon disciple grants the half-dragon template to someone who wasn’t born with it. Dragon disciple isn’t a great class, but it would work here, and it’s arguably a better option than taking half-dragon’s (or draconic’s) level adjustment. There are a lot of homebrew improvements to dragon disciple out there, too.