Kabila has backed ruling party candidate Emmanuel Ramazani Shadary, who is under European Union sanctions for his role in a crackdown on people protesting the election delay when he was interior minister.
"There's no spirit of revenge", said Tshisekedi.
The election in the DRC had been more than two years in the making.
Anti-riot police with water cannon and armored vehicles are outside Congo's electoral commission ahead of the announcement of the first results of the presidential election.
Kabila's choice of successor fuelled accusations that the long-time leader - concerned about possible retribution - would use Shadary to protect his interests after the vote.
"These people have negotiated with Kabila so that he can stay in power".
Tshisekedi's party has traditionally been strong in Kinshasa and the central Kasai region.
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Leading opposition candidate Martin Fayulu has urged the electoral commission to announce the true results as quickly as possible and warned it not to "play with fire, it is very risky".
Vital Kamerhe, another opposition leader, also abandoned the agreement and joined forces with Tshisekedi.
On Thursday, Tshisekedi paid his respects to Kabila, whom he described as "an important partner ... in democratic transition in our country".
Congo's powerful Catholic Church has upped the stakes by saying it found a clear victor from data compiled by its 40,000 observers deployed to all polling stations. Some said they would be happy as long as Fayulu or Tshisekedi won, while recalling the violence that followed past disputed elections. The work of the electoral commission must be respected, he said, but he did not know whether the FCC were planning to file a complaint against the results.
Fayulu called on the religious body for support and to proclaim him the victor. "(.) The Congolese people will never accept that his victory be stolen".
Shadary's campaign and CENI accused the bishops of violating Congolese law and risking a popular uprising. The results were originally scheduled to be released on Sunday, Jan 6.
The Catholic Church, who had the largest number of election observers deployed, has warned they will reject any results they believe to be fraudulent. A body representing Congo's Protestant leadership and SYMOCEL co-signed the document.
Current President Joseph Kabila is stepping down after 18 years in office. "At least initially, Tshisekedi will be dependent on the political favor of Kabila, who seeks immunity from prosecution and protection for his family's substantial business interests".