InstallAndRun.cs

using UnityEngine;
using UnityEditor;
using System.Collections;
using System.Diagnostics;

public class InstallAndRun : EditorWindow
{
    [MenuItem("Android/Install and Run")]
    public static void DoInstallAndRun()
    {
        string adbPath = "[AndroidSdkPath]\\platform-tools\\adb.exe"; // AndroidSdkPathは環境に合わせて変更
        string apkPath = EditorUtility.OpenFilePanel("Select apk file", "", "apk"); //apkファイルの選択ダイアログを開く

        if( string.IsNullOrEmpty(apkPath) )
        {
            return;
        }

        //インストールプロセスを実行: adb install -r apk
        Process installProcess = new Process();
        installProcess.StartInfo.FileName = adbPath;
        installProcess.StartInfo.Arguments = "install -r " + apkPath;

        installProcess.Start();

        installProcess.WaitForExit();

        // 起動プロセスを実行: adb shell am start -n YourActivity
        Process runProcess = new Process();
        runProcess.StartInfo.FileName = adbPath;
        runProcess.StartInfo.Arguments = "shell am start -n " + Application.bundleIdentifier + "/com.unity3d.player.UnityPlayerActivity";

        runProcess.Start();
    }
}