V + 을까 하다 is used to show that you are thinking about or contemplating doing something:
내년에 석사 과정을 할까 해요.
I am thinking of doing a master’s degree next year.
겨울 방학 동안에 태국을 여행할까 해서 돈을 저축하기 시작해야 돼요.
During the winter holiday I am thinking about travelling around Thailand, so I should start saving money.
한국을 떠날 때 여자친구와 브라질로 이사할까 해요.
When we leave Korea, my girlfriend and I are considering moving to Brazil.
일본어를 배울까 하는데 쓰기는 너무 힘든 것 같아요.
I am thinking about learning Japanese but it looks very difficult to write.
V + 을까 하다가 extends the idea by adding 다가 to show that an action is in some way interrupted:
기차를 탈까 하다가 표 값을 알고나서 버스를 타기로 했어요.
I considered taking the train but when I found out the ticket price I decided to take the bus.
탕수육을 주문할까 하다가 짜장면을 주문하기로 했어요.
I was thinking of ordering tangsuyuk but then I decided to order jjajangmyeon.
1주년 기념일을 축하하기 위해 여자친구와 토쿄에 갈까 하다가 시간이 없어서 제주도에 가기로 했어요.
To celebrate our first year anniversary, my girlfriend and I wanted to go to Tokyo but we didn’t have time so we decided to go to Jeju Island.