The correct option is
A Please do not doubt my capability.
The word, 'doubt' is an abstract noun which means a feeling of uncertainty about the truth, reality or nature of something. The associated verb is, 'doubt' which means to be uncertain about or to hesitate to believe or to consider something unlikely. Hence, option A is correct.
This sentence, being an imperative, should use the base form of the verb i.e doubt. However, in options B, C and D, different tenses of the verb, 'doubt' are used and hence all of them are grammatically incorrect.
Optio B: The word, 'doubting' is the present continuous form of the verb, 'doubt'.
Option C: The word, 'doubted' is the past tense form of the verb, 'doubt'.
Option D: The word, 'doubts' is the present tense form of the verb, 'doubt'.
Hence, options B, C and D are incorrect.