How should 'beseech' be used in a sentence?

To directly answer your question, your first two sentences are ok but your third one does not work:

1. RIGHT: I beseech you to help me find Magrathea.
2. RIGHT: I beseech your help in finding The Silver Bail of Peace.
3. WRONG: I beseech to you help in saving the universe.

Your first sentence uses beseech according to OED sense 3d, “To supplicate, entreat, implore a person to do a thing.”

Your second uses it in OED sense 2, “To beg earnestly for, entreat (a thing).”

But your third I cannot parse into anything sensible.

Background

I would generally advise a non-native speaker to avoid beseech altogether, because it is virtually unused today per this ngram:

And where it is used is almost in very limited contexts.

The only “modern” context that comes to mind is from the musical Godspell, which has the song “We Beseech Thee” in it, beginning this way:

Father, hear thy children's call
Humbly at thy feet we fall
Prodigals confessing all
We beseech thee, hear us!
We thy call have disobeyed
Into paths of sin have strayed
And repentence have delayed
We beseech thee, hear us!
Come sing about Love!
That made us first to be
Come sing about Love!
That made the stone and tree
Come sing about Love!
That draws us lovingly
We beseech thee, hear us!

There’s more like that, but you get the idea. Once you’ve got thee and thy in there, nobody would ever mistake it for contemporary speech.

So beseech was much more popular during the first part of the 18^th century than it is here at the start of the 21^st. The mild resurgence you see at the right end of the graph above is likely due to intentional archaism. In his Book of the New Sun published 1980–83, Gene Wolfe writes:

• The bells began again, ringing in some campanile not far away, calling the few who retained their faith to beseech the coming of the New Sun, though it was very early still, the old sun had hardly dropped Urth's veil from his face, and save for the bells the village lay silent.
• The beseeching look remained, and he spoke again.
• I spoke to him as I might have to some favorite cat upon returning from a journey, and he seemed to welcome me as a cat might, pressing his furry body to the unseen wall and mewing, regarding me with beseeching eyes.

Even Tolkien eschewed beseech in The Hobbit and The Lord of the Rings, using it only in the posthumously published Silmarillion alone, which was intended to have an archaic air about it:

• Who grieved at their long sundering from the Teleri, and besought him to bring them to Aman, if they would come.
• There they built ships, and set sail into the uttermost West upon Turgon’s errand, seeking for Valinor, to ask for pardon and aid of the Valar; and they besought the birds of the sea to guide them.
• And when Turgon heard of this he sent again his messengers to Sirion’s mouths, and besought the aid of Círdan the Shipwright.
• He named himself Wildman of the Woods, and they besought him to come and dwell with them; but he said that he had an errand yet unachieved, to seek Finduilas, Orodreth’s daughter of Nargothrond.
• He took therefore a new name, Turambar, which in the High-elven speech signified Master of Doom; and he besought the woodmen to forget that he was a stranger among them or ever bore any other name.
• They came upon Morwen by the banks of Sirion, and Mablung besought her to return to Menegroth; but she was fey, and would not be persuaded.
• Then Mîm in great fear besought Húrin to take what he would, but to spare his life; but Húrin gave no heed to his prayer, and slew him there before the doors of Nargothrond.
• Therefore I am minded to try that counsel which our forefather Eärendil took of old, to sail into the West, be there ban or no, and to speak to the Valar, even to Manwë himself, if may be, and beseech his aid ere all is lost.’

If you insist on using it, I would try to limit it formal writing. Even then I would restrict it to the sorts of uses shown above by those two 20^th-century writers.

This is not a verb I would ever encourage non-native speakers to use casually.

Beseech can generally take the place of he word ask or beg.

So, you would say:

I beseech you to help me.
I beg you to help me.
I ask you to help me.

It is typically (but not always) followed by a personal pronoun. Here are several examples of its usage in sentences.

As tchrist points out, It is highly antiquated. And, as such is always found in the dialogue of period fiction or video games.

As such, it is frequently partnered with thee and thou. But, this is because the most common usage is in liturgical language. As has been discussed on this site before, God is usually referred to as thou (to engender a sense of intimacy).

If beseeching a King, you might be more proper as the formal second person singular.