Slowly. Fortunately, for computer-controlled logging the samplerate requirements go somewhat down, allowing you to measure the filter response with no-more-so-expensive remote-controllable DMM.
A Oscilloscope with either memory or a long-exposure picture can be used, too.
Excitation can be done several ways, get a fungen with a low range, you might build a nice slow oscillator for example, use a programmable power supply, something like a HP59501A for computer control.
Also, you could hookup a I2C DAC/RDAC and ADC to arduino, should still be fast enough.