After the surgical removal of the melanoma, will I need a skin graft?

Maybe. The reconstruction after excision of a melanoma depends on the size and depth of the lesion as well as the location. Small lesions may be closed primarily while other lesions may need a local flap to close them. For larger defects, one may need a skin graft.
Depends. This depends on the location of the melanoma and the availability of local tissue to cover the defect.
Depends... The amount of skin that needs to be removed for a melanoma depends largely on the depth of the melanoma. In general, most shallow melanomas can be removed with 0.5-1cm margins and these can be easily closed side to side (resulting in a line scar) without doing a skin graft. In some locations, a flap or skin graft may be needed to close the area.
Not Usually. The amount of skin needing to be removed depends on how early a stage the melanoma was caught in. If the melanoma was early stage and the melanoma was removed completely and it was an average mole size lesion then the amount of skin needing to be excised would be modest. And if clean around the margins, then it could simply be closed without grafting otherwise it might require a graft.
Depends... The size of melanoma excision depends on the depth of the lesion and other criteria. If the excision is large and in an area that is difficult to close primarily, such as on the foot or the scalp, a graft will be necessary to close the surgical wound.