1) The first player is asked to recite a few lines concerning the death of Priam and the grief borne (a synonym?) by his wife Hecuba.H wonders at (?)the player’s skill at (?) conveying sorrow. Actually, he wonders how he could convey such a sorrow for a person he didn’t even know.

2) He wonders (how I can replace it?) who regards him as a villain when he has no courage to act.
3) He cricizes (reproves) himself for not having taken action yet and calls himself a dull, a coward, a muddy-mettled rascal (a rascal whose mettle is made of mud??), a dreamy and far-away person, and a pigeon-livered. (The gentleness of pigeons was thought to be because they had no gall in their liver). Gall was supposed to be the cause of that kind of temperament which resented insult and injury.
4) He realizes that his anger doesn’t achieve anything practical other than the unpacking of his heart (Is there a better word??)
5) The proof of his sanity is how carefully he devised plans to prepare for his revenge (or to avenge his father’s murder/revenge himself on but not for??) Can you say to take his revenge on Claudius for his father’s murder?? He could easily kill C while he is at prayer but he restrains himself so that there is no chance of Claudius’s entering heaven (no chance for C to enter heaven??)

Can you check if my sentences are correct?

1) The first player is asked to recite a few lines concerning the death of Priam and the grief borne (a synonym?) by his wife Hecuba.H wonders at (?)the player’s skill at (?) conveying sorrow. Actually, he wonders how he could convey such a sorrow for a person he didn’t even know.

"...the grief endured by ..." Your last sentence is good; use it.

2) He wonders (how I can replace it?) who regards him as a villain when he has no courage to act.
Here are some synonyms for "wonder": doubt, ponder, ask oneself, be curious, be inquisitive, conjecture, disbelieve, inquire, meditate, puzzle, query, question, speculate, think -- Which do you think would work well?

3) He criticizes himself for not having taken action yet and calls himself a dull, a coward, a muddy-mettled rascal (a rascal whose mettle is made of mud??), a dreamy and far-away person, and a pigeon-livered. What are you supposed to do with this? It seems fine as it is.

4) He realizes that his anger doesn’t achieve anything practical other than the unpacking of his heart (Is there a better word??)
http://thesaurus.reference.com/browse/unpack <~~Look through the synonyms here. There are several that would work.

5) The proof of his sanity is how carefully he devised plans to take his revenge on Claudius for murdering his father. He could easily kill Claudius while he is at prayer, but he restrains himself so that there is no chance of Claudius’s entering heaven. <~~Note the changes I made. Basically, it's OK as you originally had it.