How does the relationship between Desdemona and Othello deteriorates. How are both Iago and Emila responsible ?

Iago, of course, is primarily responsible.  He calls
attention to Cassio and Desdemona talking together innocently and suggests that
something more is going on between them.  He tells Othello that even though he does not
want to speak his thoughts, he does think that Othello might watch his wife with Cassio
a bit more closely.  He suggests that Othello does not really know how Venetian women
act, and reminds Othello that Desdemona deceived her father and is therefore capable of
deception.  He then clinches his argument with the "ocular proof" of Desdemona's
 handkerchief, telling Othello that he saw Cassio wiping his beard with it.  For the
next proof, he mentions that he overheard Cassio talking in his sleep, saying "Sweet
Desdemona, let us hide our love."  So, Iago is responsible for planting the seeds and
manipulating Othello to think Desdemona is guilty of an
affair.

Emilia is only slightly, if any, culpable.  Emilia
sees that Desdemona's handkerchief has fallen to the floor when Desdemona and Othello
were talking.  Emilia picks it up and gives it to Iago, because Iago had been pressuring
her to steal it.  To Emilia, the handkerchief means nothing.  She doesn't even want to
give it to Iago until he tells her what he is going to do with it.  But he snatches it
away from her and plants it in Cassio's lodging, so that it will seem as if Desdemona
put it there.

Later, when Desdemona is wondering where her
handkerchief is, Emilia remains silent.  Emilia does not know what her husband has done.
 She has no idea that Iago is using the handkerchief to frame Desdemona and
Cassio.