How is Hamlet portrayed as weak and indecisive?im writing research paper about the character of Hamlet. He is known to be weak and indecisive. how...

Hamlet is generally portrayed as indecisive, a man who does not act.  Lots of things do support that portrayal; I'll list a few for you to get started on your paper.

First, Hamlet resolves, after  his meeting with the Ghost (his father) to do nothing but seek the revenge his father asks of him (Iv). He says:

"...from the table of my memory
I'll wipe away all trivial fond records,
All saws of books, all forms, all pressures past,
That youth and observation copied there;
And thy commandment all alone shall live
Within the book and volume of my brain,
Unmix'd with baser matter...."

This is a strong statement--literally nothing else will occupy his time or thoughts or energies except this one thing.  As soon as his friends join him, he asks them not to "let on" if he should start to act a little crazy.  Not a particularly aggressive plan for revenge.

Some time passes; and, judging by some of the events and comments we see and hear, apparently Hamlet has enacted his "mad" plan. What we do not see is any effort at revenge.  And this continues, really, for the duration of the play.

Hamlet has begun to doubt the Ghost, wondering if he can really trust what it told him; so he hatches a plan to "catch the conscience of the king" by having the players enact The Murder of Gonzago.  After seeing Claudius' guilty reaction, Hamlet is once again resolved to avenge his father's death.

He has the perfect opportunity in IIIiii, immediately following the play.  He discovers Claudius in a moment of apparent confession, and he could easily have done the deed.  He does not. 

This is the pattern throughout the work.  Hamlet does get his revenge; however, his indecisiveness eventually gets nearly everyone he cares about killed, as well.

The two suicide speeches are other examples of his indecisiveness--"to be or not to be" is the epitome of that quality.

If this is the required focus of your writing, these ideas should get you started.  If you have some latitude with your topic, I'd like to suggest a case can be made for Hamlet as a man of action, not inaction.  In other words, every time he appears not to decide, he is, in fact deciding.  (He can act to kill Claudius; instead, he chooses to act by letting Claudius live so he can kill him when he's not apparently confessing his sins, for that's a better death than he deserves.) Just a thought.  Happy writing!