Corporate jet pilot reported receiving a low altitude alert from ATC during descent. Pilot climbed and continued inbound.

Narrative

I had executed a missed approach at ADS in low IMC conditions. ATC reported that one aircraft before me had landed at ADS; so I wanted to attempt to land there. I planned to divert to DAL if needed. After missing the approach; the controller at Lone Star asked me what my intentions were. I told him that I wanted to divert to DAL; but needed some time to set up for an approach at DAL. He stated that if I needed more than 2-3 minutes that he'd need to vector me. I told him that I'd need at least 5 minutes. I knew that there was a lot of traffic due to the low IMC conditions and felt very rushed. I was given a climb to 4000' and a heading. I climbed and engaged the autopilot. I listened to the ATIS and was handed off to the next Approach Controller. I was told to expect the ILS 13L. I started to program the FMS and also run checklists. I was given a descent to 3000' and descended. I thought I had engaged to autopilot. I was running a checklist and suddenly the Controller told me to climb immediately which I complied with. Although I thought I had engaged the autopilot; it apparently was not engaged allowing the aircraft to descend while I ran a checklist. I was then vectored onto the final approach course and landed 13L DAL.I was rushing to get the FMS programed pulling up approach plates and running checklists. In retrospect I should have asked for a delaying vector; slowed way down and got everything set up for the approach in a slower manner. Also; I know better that any time an input is made to the flight guidance panel; it must be verified on the primary flight display. I have learned a valuable lesson about slowing down and to not feel pressured just because it is a very busy terminal area. I will never make this mistake again.