In this work, we focus on improving the robot's dexterous capability by exploiting visual sensing and adaptive force control. TeachNet, a vision-based teleoperation learning framework, is exploited to map human hand postures to a multi-fingered robot hand. We augment TeachNet, which is originally based on an imprecise kinematic mapping and position-only servoing, with a biomimetic learning-based compliance control algorithm for dexterous manipulation tasks. This compliance controller takes the mapped robotic joint angles from TeachNet as the desired goal, computes the desired joint torques. It is derived from a computational model of the biomimetic control strategy in human motor learning, which allows adapting the control variables (impedance and feedforward force) online during the execution of the reference joint angle trajectories. The simultaneous adaptation of the impedance and feedforward profiles enables the robot to interact with the environment in a compliant manner. Our approach has been verified in multiple tasks in physics simulation, i.e., grasping, opening-a-door, turning-a-cap, and touching-a-mouse, and has shown more reliable performances than the existing position control and the fixed-gain-based force control approaches.