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It does not matter where you were married and what immigration/citizenship status you have. Provided that either of you lived in NJ for at least one year, NJ holds jurisdiction for divorce, per 2A:34-10.
If you have won the DV-Lottery, then the divorce should not affect this. However, upon the divorce being completed, one wants to immediately notify USCIS of the divorce, so that the visa would be for you, only. You would not file an application (DS260) for your ex.
The procedure is complex and it is recommended to use an attorney. An uncontested divorce (where the parties are in total agreement) tends to be finalized much faster than a contested divorce (where the parties disagree). For a manual, see HERE.
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