I have the core set and really like Targaryen. I am looking on suggestions to either fill out the CS or possibly build a burn deck. Any information would be greatly appreciated.

If you want to build a burn deck, you could do worse than "Dragon Attack" (x3 in A Change of Seasons).

Dragon Attack
[Neutral Event]
Any Phase: Kneel a Dragon character to choose a character. Until the end of the phase, that character gets -2 STR and is killed if its STR is 0.

If you wanted to try a burn/shadows deck, a key card would be "The Dragonpit" (x3 in Tales from the Red Keep):

The Dragonpit (3)
[Targaryen Location]
King's Landing.
House Targaryen only.
If you have at least 1 card in Shadows, any characters controlled by opponents who do not have more cards in Shadows than you do get -1 STR.

A burn/shadows deck would work well with "Free Cities Mercernaries" (x1 in The Battle of Blackwater Bay) and "King's Landing Assassin" (x3 in Secrets and Spies):

Free Cities Mercenaries (s4) (6)
[Targaryen Character]
Army. Mercenary.
House Targaryen only. No attachments. Renown.
Response: After a character with STR 2 or lower is killed, stand Free Cities Mercenaries.

King's Landing Assassin (s2) (2)
[Targaryen Character]
Ally.
House Targaryen only.
Response: After King's Landing Assassin comes out of Shadows, choose a character with no attachments. Until the end of the phase, that character gets -2 STR and is killed if its STR is 0.

There are some ideas for you. All of them untested (by me), but they look good on paper.

If it helps, click link in my signature and write str%0 as a search phrase in "Ability like" field (also uncheck "Hide Plots" option).

Thank you for the help guys. I am new to aGoT so any information is welcome.

There are some Targ discussions here and here to check out for ideas, even if they are a bit outdated.