The growing concern surrounding climate change has gradually drawn attention to the influence of climate factors on drought occurrence. In order to effectively prevent the occurrence of drought and reasonably utilize water resources, the vegetation health index (VHI) was used to characterize drought in North China Plain (NCP) in this study. Furthermore, six climate factors: air temperature (AT), precipitation (P), evapotranspiration (ET), specific humidity (SH), soil moisture (SM), and soil temperature (ST) were selected. The pole symmetric mode decomposition (PSMD) and improved gridded trend test (IGT) were used to analyze the spatial–temporal characteristics of drought and climate factors in NCP from 1982 to 2020. By calculating the cumulative climatic factors of 0 months, 1 month, 2 months, and 3 months, the correlation between drought and the climatic factors with different cumulative scales was analyzed. The results showed that: (1) from 1982 to 2020, the drought in NCP showed a downward trend and the climate factors showed an upward trend; (2) with the increase in AT, P, ET, SH, SM, and ST, VHI showed an upward trend, and SM showed the strongest correlation with VHI; (3) the optimal cumulative lag time (CLT) for AT, P, ET, SH, SM, and ST were 1.67 months, 1.48 months, 1.95 months, 1.69 months, 0.89 months, and 1.81 months, respectively; and (4) AT was the main driving factor of drought in NCP. This study contributes to the early warning and prediction of drought events, providing a scientific basis for water management authorities in drought management and decision making, and mitigating the negative impacts of drought on socio-economic aspects.