I've seen things like You're Beautiful, Bride of the Century and I am a King; can you think of any more dramas/movies that feature twins or lookalikes? I kind of like this genre but I can't find many others similar.. Any help? :)

If I remember well, in one of the Hana Kimi versions (To the beautiful you/Korean version or the 2011 remake) the girl poses as her (twin?) brother, but you don't really get to see the brother. 

In Rooftop Prince, you have a lookalike/reincarnation story, where 3 of the main characters both appear in the Joseon & contemporary story.

In the beginning You who came from the stars, the fact that the young version of the main female character looks exactly like a young girl the male main character has met hundreds of years ago is also quite important but it's not really emphasized beyond that.
Someone Like You - the male lead falls for his fiancé's doppelganger.
Arcana - not that good of a movie but it has doppelgangers
Love Letter - lookalikes, a beautiful story
Moonlight Express - lookalikes

I know one drama with twins but it's a big spoiler
Sirius
MARS
I have a lover
I think Resurrection also features twins, but I haven't seen it.
Yep, W no Higeki was nice, not everyone's cup of tea I'm afraid though... but it is an interesting drama.
"Itsuka no kimi he" it is a boys love drama
"Fuma no kojiro" a veeeery weird drama and the twis are not even the lead characters but oh well...
The only dramas I've watched with twins in are You're Beautiful, What's Up and Ouran High School Hos Club
But you also have this list:
http://mydramalist.info/topic/1168/Titles+Related+To+Twins
MARS and King Flower are both TWdramas but they're the only ones that I remember having those specifics :)
Resurrection does focus on twin brothers and features an amazing double-duty performance by Uhm Tae Woong. A good one to check out if you like revenge thrillers.