The first thing we learn about Olivia is that she is actually a bit obsessive and fanatical. To be fanatical is to be extreme and even zealous, meaning excessively enthusiastic or persistent (Random House Dictionary). In the very first scene, we learn from Duke Orsino's servant Valentine that Olivia has declared she plans to mourn her brother's death for seven years, never showing herself in public except veiled, and shall cry every day, as we see in Valentine's lines:
The element itself, till seven years' heat,
Shall not behold her face at ample view;
But, like a cloistress, she will veiled walk
and water once a day her chamber round
With eye-offending brine. (I.i.28-32)
Since even widows only typically mourned in this time period for one to two years, seven years is extremely excessive and smacks of obsession and fanaticism, especially considering that Olivia is only a sister and not a widow. In addition, allowing one's self to mourn so excessively is extremely foolish and unhealthy, as we see Feste proclaim later in Act 1, Scene 5. Hence, from this one little speech of Valentine's, we learn that Olivia is a very emotionally driven person who can behave fanatically.
A second thing we learn about Olivia is that she is a very loyal person who tries to keep everyone who is associated with her near to her, including her family and servants. For example, she continues to open her home to her uncle Sir Toby, even though she disapproves of his alcoholic behavior. Another example is seen in the fact she became so angered by Feste's prolonged absence, as shown in Act 1, Scene 5. She was so angered that at first when she sees him again she orders that he be taken away. However, Feste quickly amuses her by pointing out that she is really the true fool as opposed to him due to her prolonged mourning over a brother whose soul she believes to be in heaven. His ability to amuse her through honest criticism shows us she feels a genuine need to have Feste in her life and wants to keep him close. They further show their closeness in their later exchange of wits concerning Sir Toby's drunkenness.