He forged a paper saying that he was single so that he could take a second wife

Firstly: 

It is unfortunate that states that are supposedly Muslim are
fighting Islam and putting pressure on Muslims with regard to their rituals
and obedience to their Lord. At the time when these states are opening the
door wide to corruption, immorality and promiscuity, we find that they are
putting severe pressure on those who want to engage in plural marriage which
Allaah has permitted. Some of theses states ban it altogether, and some
stipulate that the first wife should give consent – and what wife would
agree if it were suggested to her? – and some of them stipulate an income
that many cannot attain even though they can afford to take another wife. 

These states should fear their Lord and avoid going against
the laws of Allaah. It is not permissible for any scholar or judge to accept
this legislation. If there is some misconduct and wrongdoing on the part of
some of those who have more than one wife, you will also find much more
misconduct and wrongdoing on the part of some of those who have only one
wife, so will this lead them to ban marriage to the first wife too? 

It is strange that these laws allow that which is haraam and
do not regard it as a crime or an evil, but they forbid that which is halaal
and regard it as a crime. This man whom you asked about, even though he had
an illicit relationship with this woman before he married her, if news of
that relationship had reached the authorities they would have permitted it
so long as it was with mutual consent and the woman was of sound mind!! But
if he takes her as a wife this is what they regard as forbidden and wrong,
and they say: It is permissible for you to take her as a girlfriend but it
is not permissible for you to take her as a wife! How bad is their
judgement. 

In the face of these false laws that wage war against the
laws of Allaah, there is no sin on the Muslim if he uses tricks to get
around them. 

If a man wants to take a second wife and he produces a forged
paper in order to do the marriage contract, there is no sin on him, because
the law that is preventing him from taking a second wife is a false law, and
the Muslim is not obliged to obey it or submit to it. But the husband should
examine the effects that are likely to result from this action, because it
may lead to many bad or harmful effects. 

There is also no sin on the family of the second wife who
knew of this forged paper and gave their daughter to him in marriage;
undoubtedly this is better for them and their daughter and her husband than
remaining in an illicit relationship. 

Secondly: 

The father should not force his son into a marriage with a
woman whom he does not want, and the son should not obey him in that case,
because in this case there will never be any love, compassion and kindness
between the spouses, rather the husband may mistreat his wife because he
dislikes her, and such marriages often end in failure and the spouses go
through a lot of problems and the children suffer a great deal as a result.
 

Nevertheless, the woman has done nothing wrong for which she
deserves to be mistreated because he was forced to marry her. If he wants to
please his father then he should treat his wife kindly and give her her
rights, and accept her as a wife who has full rights to be treated kindly.
If that does not happen and he is not able to keep her in a proper manner,
then he should let her go in kindness by divorcing her and giving her her
rights in full, with no shortfall. 

If the woman wants to stay married to him and be maintained
by him, and to have him care for her child without any intimacy between
them, then it is permissible for her to accept that. The same applies if he
suggests that to her and she agrees. 

It was narrated from ‘Aa’ishah (may Allaah be pleased with
her) (concerning the verse) “And if a woman fears cruelty or desertion on
her husband’s part, there is no sin on them both if they make terms of peace
between themselves; and making peace is better. And human inner-selves are
swayed by greed. But if you do good and keep away from evil, verily, Allaah
is Ever Well‑Acquainted with what you do” [al-Nisa’ 4:128] that she
said: This was revealed concerning a woman who is married to a man, and has
been with him for a long time, then he wants to divorce her, and she says:
Do not divorce me; keep me and you have no obligation towards me. Then this
verse was revealed.

Narrated by al-Bukhaari (2318) and Muslim (3021). 

According to a report narrated by al-Bukhaari (2584)
‘Aa’ishah said: This refers to a man who sees something in his wife that he
dislikes because of old age or something else, so he wants to leave her, but
she says Keep me, and give me whatever you want. She said: There is nothing
wrong with that if they both agree. 

Ibn al-Qayyim (may Allaah have mercy on him) said: 

If a man is no longer interested in his wife and he does not
like her any more, or he is unable to give her her rights, then he may
divorce her or he may give her the choice: if she wishes she may stay with
him and have no right to a share of his time or intimacy or maintenance, or
she may have some of that, according to whatever she agrees with him. If she
agrees to that then it is binding and she does not have the right to ask for
it after agreeing to something.  

This is in accordance with the Sunnah and it is the correct
view; no other view is justified. The view of those who say that her rights
may be restored and she may recant this whenever she wants is wrong, because
this is a deal where one party has given up something in return for
something else. Allaah has called it a sulh (amicable settlement), and it is
binding, as is any settlement concerning rights and business deals. If she
was able to demand her rights after that, this would just delaying the
problem without solving it, and this is not a settlement, rather it is a
recipe for greater problems, and sharee’ah would not call for that. One of
the signs of the hypocrite is that when he makes a promise he breaks it, and
when he makes a pledge he betrays it. 

Zaad al-Ma’aad (5/152, 153). 

Shaykh al-Islam Ibn Taymiyah said something similar in
Majmoo’ Fataawa Ibn Taymiyah (32/270). 

Thirdly: 

You say in your question that he was in an illicit
relationship with her. If that means zina, then you should note that the
marriage of a zaani or zaaniyah is not valid unless they repent. This has
been discussed in the answers to questions no.
11195 and
14381. 

And Allaah knows best.