ㄹ래요 is more colloquial, and is use only for first person statements and second person question. The translation would be "I feel like..", "do you feel like...?"
먹을래요? Do you feel like eating? ~Wanna eat?
수영할래요. I feel like swiming
ㄹ까요 Is more formal, and has two different uses>
1.-Like a tentative question "Should I...?" "Should we...?", used only with action verbs (Go, run, sleep etc) in the present tense