A pilot reported they flew off course while departing on a SID and received a Low Altitude warning and expedited climb instruction from ATC.

Narrative

I departed ZZZ Runway XX on an IFR flight plan via the ZZZ SID. As I was turning left to intercept the ZZZ vortac 114 radial at approximately 3;500 to 4;000 ft.; I was issued the flowing climb instruction; 'expedite thru 6;000 climb and maintain one-one thousand.' I acknowledged this clearance and immediately began an intense visual scan outside the aircraft for traffic because of the Controller's unusual request for an expedited climb. I did not properly include my navigation receiver as part of my scan as I was now concentrating outside the plane where VMC prevailed at that time. Less than a minute later; the Controller issued a 'low altitude alert' to which I responded that I was in VMC and had terrain in sight. I was not aware at this time that I was off course from the radial specified in the SID. Approximately 1 minute later; as I was climbing thru 6;000 ft.; I was subsequently issued a radar Vector to 'ZZZZZ intersection; flight planned route;' Post flight review of 'Flight Aware' tracking showed my flight east of the ZZZ 114 radial and return after the low altitude from the Controller. I should have remained focused on tracking the ZZZ 114 radial after departure and not allowed the unusual expeditious climb request from the Controller distract me away from my primary function of navigating. Also; if the Controller witnessed my flight deviating left off course towards terrain; it would have been better for her to have notified me of a course error rather than issue an expedited climb instruction.