You can use the new YouTube Data API v3
if you retrieve the video, the statistics part contains the viewCount:
from the doc:
https://developers.google.com/youtube/v3/docs/videos#resource
statistics.viewCount / The number of times the video has been viewed.
You can retrieve this info in the client side, or in the server side using some of the client libraries:
https://developers.google.com/youtube/v3/libraries
And you can test the API call from the doc:
https://developers.google.com/youtube/v3/docs/videos/list
Sample:
Request:
GET https://www.googleapis.com/youtube/v3/videos?part=statistics&id=Q5mHPo2yDG8&key={YOUR_API_KEY}
Authorization:  Bearer ya29.AHES6ZSCT9BmIXJmjHlRlKMmVCU22UQzBPRuxzD7Zg_09hsG
X-JavaScript-User-Agent:  Google APIs Explorer
Response:
200 OK
- Show headers -
{
 "kind": "youtube#videoListResponse",
 "etag": "\"g-RLCMLrfPIk8n3AxYYPPliWWoo/dZ8K81pnD1mOCFyHQkjZNynHpYo\"",
 "pageInfo": {
  "totalResults": 1,
  "resultsPerPage": 1
 },
 "items": [
  {
   "id": "Q5mHPo2yDG8",
   "kind": "youtube#video",
   "etag": "\"g-RLCMLrfPIk8n3AxYYPPliWWoo/4NA7C24hM5mprqQ3sBwI5Lo9vZE\"",
   "statistics": {
    "viewCount": "36575966",
    "likeCount": "127569",
    "dislikeCount": "5715",
    "favoriteCount": "0",
    "commentCount": "20317"
   }
  }
 ]
}