To be self-indulgent is to be obsessed with satisfying "one's own desires, passions, whims ... especially without any restraint" (Random House Dictionary). In other words, one is so caught up in one's own wants and thoughts that the person is incapable of paying attention to anything else, particularly others' own wants and thoughts. The very first scene of the play certainly clearly portrays Duke Orsino as being self-indulgent. He's so obsessed with his painful feelings of love for Olivia due to her rejection that he can't think of anything else.
One good example is that after he commands the musicians to stop playing, his other attendants suggest he go hunting; still, all he thinks about is his aching heart. For example, after Curio suggests Orsino go hunting after "the hart," meaning male deer, all Orsino can do is make a pun out of the word hart to reflect again on his own lovesick sorrows, as we see in his reply:
Why, so I do, the noblest that I have:
O, when mine eyes did see Olivia first,
Methought she purg'd the air of pestilence! (I.i.16, 18, 19-21)
In these lines, Orsino has interpreted the word hart, which is a male deer, as the word heart, which is of course the human organ linked to existence and even figuratively associated with emotions. Hence, what Orsino is saying in these lines is that he already is hunting "heart"; he is hunting the desires of his own noble heart. He then continues to rave about Olivia's beauty, saying that her beauty seemed so pure to him when he first saw her that he could imagine her beauty being able to purge the world of diseases. Thus, since Orsino's attendant is trying to distract him from is lovesickness, but his only reply dwells even further on his own lovesickness, it shows us that he is obsessing on his own emotional state and refusing to pay attention to anyone or anything else, which is indeed a perfect description of self-indulgence.