How does Renji become stronger than Rukia?
When Rukia stepped into the world of the living, she had to inhabit a "Gigai" (empty body) to be able to interact with the living. That body was provided by Kisuke Urahara. Hidden inside that Gigai was one of his inventions (the Hōgyoku), which by being placed in the same body as a Shinigami would be destroyed, or so he planned. After Rukia gifting her Shinigami power to Ichigo, her powers failed to return, and the Hōgyoku was what was blocking Rikia's powers. Along with the tower she was held in during her wait for execution, she was left with no power. Shortly after she was saved, the Hōgyoku was stolen, and her power began to very slowly return.

In the later episodes, when she has recovered, you can clearly see her strength after she has enough power to call forth her Zanpakutō with its overwhelming ability to freeze enemies.

Just as Ichigo has gotten stronger from numerous fights with Hollow and Shinigami, so has Renji from their fights and his own determination to save Rukia from her execution and later protect her from further harm. Also, his power was boosted considerably with him mastering his Bankai at the same time Ichigo was unlocking his own.

Plus, all Shinigami have their power capped/suppressed when visiting the human world, or they would seriously damage the living.