I'm pretty sure that Climbing is based on Agility, and then boosted by the availability of rope and grappling hook. HOWEVER, if you didn't build the character with A LOT of Agility to begin with, expect a lot of failing. During character build, Attribute point allocations cost 1 for 1 up to 29. At 30, Attributes cost 2 for 1. Then 3 for 1 at 40. You've only got 97 EPs max to spend across six Attributes. Just so a character isn't godawful in one or two attributes, you can NOT afford to seriously boost more than one Attribute. An Agility of, say, 29 (highest 1-for-1 value) is actually pretty poor = LOTS of fails. It isn't until the Attribute is up towards 40 that the Attribute becomes dependable.
As long as you are NOT hardcore, the surest way to succeed is to Save prior to the attempt. There's _always_ at least a small chance of success. Save. Fail. Load. Repeat as often as necessary until you succeed.