Simply put, Juliet uses Romeo's dagger because it is there. When Juliet awakens from the effects of the potion she expects Romeo to be there and for the two of them to be able to walk off into the sunset together. However, by the time she comes to Romeo has drunk the poison and is dead. When she cannot glean enough poison from the vial or from Romeo's lips to kill herself she grabs the nearest thing at hand, Romeo's dagger.
Juliet is also in a bit of a hurry as she has heard the guards at the tomb door and knows she doesn't have much time before they will enter and possibly take her away from her love. So she goes for the quickest and simplest option that is provided for her.
Before I go into why Juliet killed herself, we need to explain why Romeo did in the first place!
Romeo and Juliet were extremely eager to get married, but they couldn't as their families hated each other, and would forbid such activity. But, being in their youth and being blinded by love, they come to desperate measures in order to get married.
Romeo and Juliet ask a religious Friar friend, Lawrence to come and wed them secretly. Reluctantly, Friar Lawrence agrees and engages them together. Now, all Romeo and Juliet have to do is elope to somewhere where their families would never find them, but it's not as simple as packing their things and leaving. Romeo leaves for the night to think things through.
Juliet, being impatient (and with a wedding coming up, as her father has forced her to marry Paris, whom she does not want to marry) decides to ask Friar Lawrence to drug her with a certain potion that would put her into such a deep sleep that it would appear if she were to be dead. That way, her family (the Capulet's) would think that she would be dead. Friar Lawrence agreed, and said that he would send a letter to Romeo explaining what is happening.
Juliet takes the 'deep sleep' potion and falls onto her bed. Her family comes up to her bedroom and suspects that she is dead, and give her a funeral. The word goes around that Juliet had died, when really she was only in a deep sleep.
Unfortunately, the letter Friar Lawrence had sent to Romeo never made it to him. Instead, it got lost and was never seen again. Romeo heard of Juliet's random and unsuspecting death, and was naturally in shock. He went to the apothecary (a place where medicines and potions are sold) and bought a lethal potion that would kill him quite quickly.
Romeo went to Juliet's temporary tomb, gave his final solioquy (speech) (after fighting off Paris) and drank the potion, dying almost instantly right next to Juliet's tomb. Ironically, Juliet wakes up only seconds later to find Romeo dead on the ground from a potion. She, also in shock reacts unreasonably to the death of Romeo and tries to lick all the lethal potion she can off Romeo's lips. Unfortunately for her, it was not enough to end her woes, so she picked up Romeo's dagger and stabbed herself, killing her almost instantly.
If you don't understand why Juliet killed herself, it is because of the shock and the devastation she got from seeing her love dead on the ground from drinking lethal potion. She, being only thirteen did not think things through and ultimately killed herself because of all the negative thoughts and emotions she got from seeing her one love, Romeo dead.
When everything ends up like this (where the protagonist/s kill themselves or die in the end) it is called a tragedy. Romeo and Juliet may be a love tale, but it is ultimately, and unfortunately a very sad tragedy.
Hopefully I've answered your question :).