Being a Time Lord is not specifically a genetic trait. The title and distinction belongs solely to those beings from the world of Gallifrey who have been indoctrinated, trained and introduced to the Time Vortex via a particular set of rituals, instruction and guidance.
Currently there are a few individuals who possess DNA from the world of Gallifrey including: The Doctor, The Master, River Song, the Doctor's Daughter (a female DNA construct of the Doctor whose whereabouts are currently unknown) and the renegade Time Lord from far ago in Doctor Who history which is sure to appear as soon as I finish writing this.
Education is one of the primary elements which defines a Time Lord. The other, is being linked to a TARDIS.
One of the major institutions of the Time Lords was the Time Lord Academy. It was split up into Chapters, each of which was identified by its distinctive colours. (TV: The Deadly Assassin) The subjects at the academy ranged from the study of Cosmic Science (TV: Terror of the Autons) to Veteran and Vintage Vehicles (TV: The Pirate Planet) to Gallifreyan flutterwings (TV: The Pirate Planet).
Children began instruction at the Time Lord Academy, at the age of 8, in a special ceremony. The Gallifreyans would be forced to look into the Untempered Schism, which showed the entirety of the Time Vortex and the power that the Time Lords had. The Gallifreyans subjected to its terrifying effects would react differently: "Some would be inspired, some would run away, and some would go mad." (TV: The Sound of Drums) They would then spend "centuries" studying at the Academy. (COMIC: Mortal Beloved)
Only the Doctor and the Master have had the proper "education" to be considered Time Lords. All of the others have some DNA and some affiliation with the Time Vortex and/or the TARDIS but none of the specialized awareness unique to the Time Lords of Gallifrey.
Since the Gallifreyans are locked beyond the Universe in the Time War, there are unlikely to be any new Time Lords or available TARDISs any time soon.
See Also: Can there be any more NEW Time Lords?