Praise be to Allah.

Firstly: 

First of all, this man has to repent greatly to Allah, may He be exalted, from the grave evil deeds and major sins that he has committed, and he must be sincere towards Allah in repenting, regret what he has done, ask a great deal for forgiveness, and do a lot of righteous deeds.

Please see also the answers to questions no. 14289 and 128111. 

Secondly: 

In the answer to question no. 169633 it was explained that in order for repentance that has to do with transgressions against the rights of other people to be valid, it is essential to restore people’s rights or ask for forgiveness. 

So if a person has stolen something, he has to return the property to the person from whom it was stolen, or to his heirs if that person has died. If he does not know who that person was or has no way of contacting him, then he should give that wealth in charity on his behalf, on the basis that if (the original rightful owner) ever comes along, he will give him the choice between approving of the act of charity or having the wealth given back to him. 

If the value of the stolen property is not known, then he should look to what he thinks will most likely discharge his obligation. 

Please see also the answers to questions no. 83099 and 142235 for further details. 

Thirdly: 

As he has previously done Hajj on his own behalf, it is permissible for him to do Hajj on behalf of his paternal aunt, if she has died or if she is still alive but she is elderly and unable to do Hajj, or she has a sickness from which there is no hope of recovery, as a result of which she is unable to travel to Makkah and do the rituals of Hajj. 

However, if she has a temporary excuse that is preventing her from doing Hajj, but there is the hope that this excuse will no longer apply in the future, such as if she is sick with an illness from which it is possible to recover, by Allah’s leave, then it is not permissible for him to do Hajj on her behalf, because she comes under the ruling of those who are able to do it. 

Please see the answer to question no. 11407

But it is preferable for everyone, especially one who is in such a situation of carelessness and wrongdoing himself, to do Hajj on his own behalf, because the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever comes on pilgrimage to this House and does not utter any obscene speech or do any evil deed, will go back (free of sin) as his mother bore him.” Narrated by al-Bukhaari (1449) and Muslim (1350). It was also narrated in a saheeh report that he (blessings and peace of Allah be upon him) said: “Follow one Hajj or ‘umrah with another, for they eliminate poverty and sin as the bellows eliminates the dross of iron, gold and silver, and an accepted Hajj brings no less a reward than Paradise.” Narrated by al-Bukhaari (810) and elsewhere; classed as saheeh by al-Albaani in Mishkaat al-Masaabeeh (2524). 

As that is the case, then it is better for a person in such a situation to turn over a new leaf with his Lord, and strive to wash away whatever burden he has of sin and error. 

What he must do is strive to make sure that this Hajj of his comes after repenting and turning back to Allah, and that it is done with halaal wealth concerning which there is no suspicion. So he should start first of all by returning the stolen property and apologising for that and asking the owners to forgive him, then he should do Hajj with his own halaal wealth. 

Ibn ‘Abd al-Barr (may Allah have mercy on him) said: 

With regard to the accepted Hajj, it was said that it is that in which there is no showing off, seeking reputation, obscene speech or evil deeds, and it is done with halaal wealth. End quote.

At-Tamheed (22/39) 

But if he does Hajj – whether that is on behalf of himself or on behalf of his paternal aunt – and he had not yet repented from his sins or restored people’s rights, then in this case there is the fear of a bad end or that his deeds may be rendered invalid – Allah forbid. 

If you do Hajj with money from impure sources, then you have not done Hajj but perhaps the camel you rode did Hajj! 

Please see also the answer to question no. 131552

And Allah knows best.