After the Rajasuya by Yudhistira, Duryodhana also tries to perform it with the help of Karna.
What is the intention of Duryodhana in performing Rajasuya? Is it mentioned anywhere in Mahabharata?
Yes, it is mentioned in the Mahabharata. (In the GhoshYatra Parv, Chapter 253 to 256).
After the exile of Pandavas, Duryodhana planned to perform a Rajasuya yagy similar to Yudhishthira, because he wanted to become a Samrat (Emperor). For this, he asked Karna to fight all the kingdoms who opposed him and collect revenue for him, like the same way done by four Pandavs for Yudhishthira.
But when he tried to conduct the Yagy, he was stopped by the Priests (Vyas and other high Priests), saying that there could only be a single Samrat at one time, and as Yudhishthira is still alive, even in his exile he is still a legal Samrat, so he cannot perform Rajasuya.
From Chapter 253, (translated by KM Ganguli)
Dhritarashtra's son summoned the priest, and spake unto him these words, 'Do thou duly and in proper order celebrate for me that best of sacrifices, the Rajasuya furnished with excellent Dakshinas.' Thus accosted, that best of Brahmanas spake unto the king, saying, 'O foremost of the Kauravas, while Yudhishthira is living, that best of sacrifices cannot be performed in thy family, O Prince of kings! Further, O monarch, thy father Dhritarashtra, endued with long life, liveth. For this reason also, O best of kings, this sacrifice cannot be undertaken by thee.
Instead, the Priests performed one another Yagy for him (Vaishnava Yagy), which is almost similar to Rajasuya but not Rajasuya (Lesser recognition than Rajasuya and does not makes you Samrat).
There is, O lord, another great sacrifice, resembling the Rajasuya. Do thou, O foremost of kings, celebrate that sacrifice. Listen to these words of mine. All these rulers of the earth, who have, O king, become tributary to thee, will pay thee tribute in gold, both pure and impure. Of that gold, do thou, O best of monarchs, now make the (sacrificial) plough, and do thou, O Bharata, plough the sacrificial compound with it. At that spot, let there commence, O foremost of kings, with due rites, and without any disturbance the sacrifice, sanctified with mantras abounding in edibles. The name of that sacrifice worthy of virtuous persons, is Vaishnava. No person save the ancient Vishnu hath performed it before. This mighty sacrifice vies with that best of sacrifices--the Rajasuya itself. And, further, it liketh us--and it is also for thy welfare (to celebrate it).