Neither of the two Dragon types available in HeartGold and SoulSilver (Dragonite and Kingdra) are very good.
Dratini is first obtained in Goldenrod City and can use Dragon Rage to 2HKO a majority of this part of the game. Unfortunately, playing Voltorb Flip for the requisite coins takes a long time and is generally inefficient. In addition, Dratini levels up slowly and doesn't have amazing matchups outside of Clair, so it won't typically be seeing a lot of use in battle. Dragonite also comes at such a high level that you're likely not going to have it until Kanto, so you'll be stuck with Dragonair for the Lance fight. Dratini is even worse if you wait for the one with ExtremeSpeed -- the grinding necessary to bring it up to speed is never worth the unique moveslot.
Horsea is available fairly late, first catchable in the Whirl Islands. It's decent against opponents like Jasmine and Clair with powerful Hydro Pumps and Draco Meteors. To get a Dragon Scale, however, is a long endeavour -- the only way to get one is at a 5% chance from wild Horsea. This time sink and the fact that Kingdra has to rely on the low-accuracy and low-PP Hydro Pump for most of its OHKOs mean that it's a suboptimal choice for playthroughs.
For further reading, I would recommend Smogon's in-game viability rankings.
