Roman Catholics have a hierarchical system within their church with cardinals, then archbishops, then bishops, then priests and then deacons (as i have shown below) but where in the Bible do we see the hierarchical system of the Roman Catholic Church?
The hierarchy of the Catholic Church strictly speaking refers to Deacons, Priests and Bishops. This is the holy ordering of the church and the Scriptural basis for it can be found at 1 Corinthians 12 and Timothy 3. The phrase "hierarchy" is non-biblical and purely descriptive being made popular in the sixth century as a means of articulating the ordering.
Archbishops, Cardinals and Popes are all Bishops (or in some very rare cases Priests or Deacons). These are honorific distinctions that clarify the role of the ordained person. Archbishops, for example, are merely Bishops who oversee a major diocese or collection of dioceses and Cardinals are merely Bishops appointed to ceremonially head Roman dioceses and elect the Bishop of Rome when the seat is vacated.
The hierarchy presented in the graphic you shared is also strictly Latin in nature. Several autonomous particular Churches have major archbishops with various unique titles and styles and many sui iuris churches have patriarchs who head them. These are fundamentally Bishops as well in the catholic ordering, but they again have honorific titles and styling that reflect their position within their particular churches. The Bishop of Rome was once styled the Patriarch of the West because of these positions but that was stopped under Benedict XVI.
The long and the short is that the Catholic Church does not have a unique or special hierarchy within Christianity than the vast majority of churches and communities that have an ordering of Deacons, Priests and Bishops. While there is certainly a strain of Christianity that is not episcopal or presbyterian in nature (i.e. having very little organization), there is Scriptural evidence for an ordering especially in Paul's letter to Timothy who articulates the offices of Bishop and Deacon clearly.
What is the biblical basis for the hierarchical system of the Catholic church?
A hierarchical system within the Catholic Church has it’s beginnings within the Scriptures itself. But should be pointed out that other denominations also have their own hierarchical systems, such as the Orthodox, Lutherans, Anglicans and others.
We can see glimmers of an organized Church in the Acts of the Apostles.
1 At this time, as the number of the disciples increased, complaints were brought against those who spoke Hebrew by those who spoke Greek; their widows, they said, were neglected in the daily administration of relief. 2 So the twelve called together the general body of the disciples, and said, It is too much that we should have to forgo preaching God’s word, and bestow our care upon tables. 3 Come then, brethren, you must find among you seven men who are well spoken of, full of the Holy Spirit and of wisdom, for us to put in charge of this business, 4 while we devote ourselves to prayer, and to the ministry of preaching. 5 This advice found favour with all the assembly; and they chose Stephen, a man who was full of faith and of the Holy Spirit, Philip, Prochorus, Nicanor, Timon, Parmenas, and Nicolas, who was a proselyte from Antioch. 6 These they presented to the apostles, who laid their hands on them with prayer. - Acts 1:1-6
Thus the Apostles saw the needs in Apostolic Times to ordain deacons, so that their Apostolic work of preaching could be more easily accomplished.
The seven men elected by the whole company of the original Christian community at Jerusalem and ordained by the Apostles, their office being chiefly to look after the poor and the common agape. The number of believers at Jerusalem had grown very rapidly, and complaints had been made that poor widows of Hellenistic Jews were neglected. The Apostles, not desiring to be drawn away from preaching and the higher spiritual ministry to care for material things, proposed to the believers to transfer such duties to suitable men, and following this suggestion the "Seven" were appointed (Acts 6:1-6). This was the first separation of an ecclesiastical, hierarchical office from the Apostolate in which up to then the ecclesiastico-religious power had been concentrated. The "seven men" were "full of the Holy Ghost" and therefore able partially to represent the Apostles in more important matters referring to the spiritual life, as is seen in the case of St. Stephen at Jerusalem, of St. Philip in Samaria, and elswhere. Nothing further is known of several of the seven deacons, namely Nicanor, Timon, and Parmenas. Philip, who is called the "Evangelist", preached with much success in Samaria (Acts 8:5 sq.), so that the two Apostles Peter and John went there later to bestow the Holy Ghost on those whom he had baptized. He also baptized the eunuch of the Queen of the Ethiopians (Acts 8:26 sqq.). According to the further testimony of the Book of the Acts (xxi, 8 sqq.) he lived later with his prophetically gifted daughters at Caesarea. - Seven Deacons
Furthermore St. Paul speaks of various gifts given to the Church in order to help in the spreading of the Gospel.
And he gave some apostles, and some prophets, and other some evangelists, and other some pastors and doctors. - Ephesians 4:11
The above Scriptural reference is token from the Douay-Rheims 1899 version of the Bible. Some translations have teachers for doctors.
St. Paul the Apostle set downs his expectations of is required of bishops.
4 To Titus my beloved son, according to the common faith, grace and peace from God the Father, and from Christ Jesus our Saviour.
5 For this cause I left thee in Crete, that thou shouldest set in order the things that are wanting, and shouldest ordain priests in every city, as I also appointed thee:
6 If any be without crime, the husband of one wife, having faithful children, not accused of riot, or unruly.
7 For a bishop must be without crime, as the steward of God: not proud, not subject to anger, not given to wine, no striker, not greedy of filthy lucre:
8 But given to hospitality, gentle, sober, just, holy, continent:
9 Embracing that faithful word which is according to doctrine, that he may be able to exhort in sound doctrine, and to convince the gainsayers. - Titus 1:4-9
3 A faithful saying: if a man desire the office of a bishop, he desireth a good work.
2 It behoveth therefore a bishop to be blameless, the husband of one wife, sober, prudent, of good behaviour, chaste, given to hospitality, a teacher,
3 Not given to wine, no striker, but modest, not quarrelsome, not covetous, but
4 One that ruleth well his own house, having his children in subjection with all chastity.
5 But if a man know not how to rule his own house, how shall he take care of the church of God?
6 Not a neophyte: lest being puffed up with pride, he fall into the judgment of the devil.
7 Moreover he must have a good testimony of them who are without: lest he fall into reproach and the snare of the devil.
8 Deacons in like manner chaste, not double tongued, not given to much wine, not greedy of filthy lucre:
9 Holding the mystery of faith in a pure conscience.
10 And let these also first be proved: and so let them minister, having no crime. - 1 Timothy 3:1-10
Again St. Paul speaks about the various ministries within the Church:
5 And there are diversities of ministries, but the same Lord;
6 And there are diversities of operations, but the same God, who worketh all in all.
28 And God indeed hath set some in the church; first apostles, secondly prophets, thirdly doctors; after that miracles; then the graces of healing, helps, governments, kinds of tongues, interpretations of speeches.
29 Are all apostles? Are all prophets? Are all doctors?
30 Are all workers of miracles? Have all the grace of healing? Do all speak with tongues? Do all interpret?
31 But be zealous for the better gifts. And I shew unto you yet a more excellent way.- 1 Corinthians 12:5-5 & 28-31
More information may be gleaned from the following articles:
Saints Timothy and Titus Saint Paul could not do it alone so he appointed helpers
As for the title of Cardinals, it did not come about for centuries later and it’s development is explained here. Their main duty is to elect the next Supreme Pontiff for Universal Catholic Church at the death or resignation of a standing pope.
The title of cardinal is more a honorific title within the Catholic Church that carries with it many duties and responsibilities. They are now also called the Princes of the Church and wear red cassocks as a sign that they should be ready to shed their blood for Christ and his Church.