What does the Church demand of a person who is married to four wives and who wants to adhere to the Christian faith and be baptized?

Answer:

The teaching of the Church about the love of husband and wife remains valid for each Catholic Christian believer. 

The married couple forms the intimate partnership of life and love established by the Creator and governed by his laws; it is rooted in the conjugal covenant, that is, in their irrevocable personal consent. Both give themselves definitely and totally to one another. They are no longer two; from now on they form one flesh. The covenant they freely contracted imposes an obligation on the spouses to preserve it as unique and indissoluble. What therefore God has joined together, let no man put asunder (Mk 10, 9).   (Catechism of the Catholic Church, n. 2364)

Please refer to theme 10: The vocation of the Christian. II. The Christian view. See also: Questions and Answers nr. 114; 86; 258; 18

On the basis of this understanding of marriage, Paragraph 1148 of The Canon Law of the Church states: 

When he receives baptism in the Catholic Church, a non-baptized man who has several non-baptized wives at the same time can retain one of them after the others have been dismissed, if it is hard for him to remain with the first one. The same is valid for a non-baptized woman who has several non-baptized husbands at the same time.  Canon 1148 §1

In the cases mentioned in §1, marriage must be contracted in legitimate form after baptism has been received, and the prescripts about mixed marriages, if necessary, and other matters required by the law are to be observed.  Canon 1148 §2

Keeping in mind the moral, social, and economic conditions of places and of persons, the local ordinary is to take care that the needs of the first wife and the others dismissed are sufficiently provided for according to the norms of justice, Christian charity, and natural equity.  Canon 1148 §3