How can I write a soliloquy like Shakespeare? Where should I start?

First of all, don't let the word 'soliloquy' scare you. It simply means a character speaking all by himself, to himself. 

Shakespeare uses soliloquies to show a character's thoughts. The subject of the soliloquy often involves a character trying to make a difficult decision. For example, Macbeth has a soliloquy in Act 5 scene 5:

To-morrow, and to-morrow, and to-morrow,
Creeps in this petty pace from day to day,
To the last syllable of recorded time;
And all our yesterdays have lighted fools
The way to dusty death. Out, out, brief candle!
Life's but a walking shadow, a poor player,
That struts and frets his hour upon the stage,
And then is heard no more. It is a tale
Told by an idiot, full of sound and fury,
Signifying nothing.

Here he shows the audience his despair. The glory, power, and fame he envisioned when he first murdered Duncan have not materialized. Instead life has lost all joy and purpose.

When you start writing a soliloquy, first pick a character who has some kind of strong feelings or a difficult decision to make. Then imagine what that character would say about the situation. Is he angry? Is he sad? Is he frightened? 

Once you decide that, write some lines that you think this character would say to express those feelings. If he is trying to make a decision, he would talk about both sides of the problem...what could happen if he made one decision, or what might be the outcome of another decision?

You haven't mentioned if you need to use Shakespearean English or how long this needs to be. But if you put yourself in the mind of the character and concentrate on what he or she would say about the current situation, you should be able to write a strong, convincing soliloquy!