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During your travels, you'll come across several pieces of dragon armor. This is where you can find them.
Belt of the Dragon. In an abandoned house (Verdistis, #19)
Gloves of the Dragon. In the Elven Burial Grounds (Dark Forest, #27)
Shield of the Dragon. In Stormfist Castle (Northern Dukedom, #2)
Heart of the Dragon (aka the Dragonheart Gem). Carried by Rak'Sheen (Imp Castle, #6)
Breastplate of the Dragon. Given by Sir Patrick (Verdistis, #14). You'll only receive the dialogue option to acquire the breastplate after King Dunatrim tells you to look for it with a wealthy family in Verdistis. Then Sir Patrick will offer it to you if you can convince Lucius to pay his debt. Of course, Lucius won't be able to pay, so in essence you'll have to pay 2500 gold to acquire the breastplate, but you'll also receive 46,000 experience and 2 reputation points, so it's money well spent.
Breastplate of the Dragonheart. Crafted by Grischa (Dwarven Halls, #5) after you've acquired the Breastplate of the Dragon and the Heart of the Dragon. The crafting will destroy the old breastplate, and the new breastplate, while potentially better, won't have anything to do with it.
Helmet of the Dragon. Given by King Dunatrim (Dwarven Halls, #2) after you've acquired the Breastplate of the Dragonheart.
Once you've collected all the parts, if you then wear them together you'll receive a point of "elemental strike."