If you want to deploy github private repo to netlify, using netlify API. Here is a step by step procedure.

1. Firstly you need to get the id of the github repo, which you want to deploy. know how to get the github id here.
2. After getting repo id, You have to get public_key & id using netlify api. know how to get the public_key here.
3. After getting public_key, Using github api https://api.github.com/repos/:owner/:repo/keys POST public_key in the body section as shown below

{
    "title":"test",
    "key":<your public key>,
    "read_only":false
}

4. After this step deploy the site using https://api.netlify.com/api/v1/sites. Use POST method by body as below

{
  "name":<your site name>, // ex: test
  "repo":{
      "provider":"github",
      "id":<your github repo id>,
      "repo":"<your username>/<your repo>",
      "private":true,
      "branch":"main",
      "deploy_key_id":<your deploy key id>
  }
}

In the above deploy_key_id is the id you got in the 2nd step. 5. Boom!! congratulations, your site has been deployed successfully.

if you found any trouble feel free to add issue