I was wondering if there was a way to multiclass into a dragon disciple on here. 

Is Dragon disciple a homebrew or previous edition class?

I assume you are talking about the 3.5 prestige class, 5e doesn't have prestige classes, I would see if draconic bloodline fits the theme you want, alternately maybe look at a dragonborn valor or blades Bard if you want a more martial take on the theme

The Draconic Bloodline Sorcerer is the 5e version of Dragon Disciple. For comparison:

Dragon Disciple:

1. Requires a nondragon race, i.e. cannot already be a half-dragon.
2. Requires the ability to cast arcane spells without preparation. Of the base classes, that means Sorcerer and Bard.
3. Requires Knowledge (Arcana) and must know the Draconic Language.
4. Gains bonus spells.
5. Choose a dragon variety at first level that determines later features.
6. Becomes more draconic in appearance. Their skin develops tiny iridescent scales, nearly invisible at first but becoming more noticeable at higher levels. As their skin thickens, a dragon disciple takes on more and more of their progenitor’s physical aspect.
7. Gains natural armor, up to a +3 bonus, which is not applied when you have an armor bonus equal to or greater than your natural armor bonus.
8. Has a d12 HD per level.
9. Gains a claw and bite attack. Gains a breath weapon based on their dragon variety. Gains ability boosts to Str, Con, and Int. Gains blindsense.
10. Dragon wings.
11. Gains the half-dragon template, granting the following. Gains an additional bonus to Str and Cha. Gains an additional natural armor bonus. Gains low-light vision and darkvision. Immune to sleep and paralysis. Immune to the energy type used by your breath weapon.

Draconic Bloodline:

1. Available to all PC races, including dragonborn.
2. Only available as a Sorcerer subclass.
3. Although not required, Arcana is a Sorcerer class skill and the Draconic Bloodline grants Draconic as a bonus language.
4. Bonus spells aren't inherent to every spellcaster, and when they are available they don't work the same way in 5e. A Sorcerer's Font of Magic is their equivalent to bonus spells in this edition, replacing bonus spells gained in previous editions by having a high spellcasting ability score or from a class feature. You also continue to gain levels in the Sorcerer base class, increasing all of your spellcasting capabilities rather than just bonus spells.
5. Choose a dragon ancestor at first level that determines later features.
6. As magic flows through your body, it causes physical traits of your dragon ancestors to emerge. Additionally, parts of your skin are covered by a thin sheen of dragon-like scales.
7. When you aren’t wearing armor, your AC equals 13 + your Dexterity modifier. (natural armor equivalent)
8. Increases hit point maximum by 1 every level.
9. Gains bonus damage to spells if the damage is the same type associated with your dragon ancestor, and you can also gain resistance to that type.
10. Dragon wings.
11. Channels a dread presence, which creates an aura around you that causes creatures to become charmed or frightened.

You can play a dragonborn to gain the breath weapon and a bonus to Str and Cha, though there is some redundancy in having two sources for knowing the draconic language and resistance if your heritage and ancestor are of the same type.