I have implemented this scenario in two ways. Not sure they are the best or not but works well for me.
In both cases you have to store the refresh token along with the email id of the user. Then you have to make a HTTP request in the below format.
POST /oauth2/v4/token HTTP/1.1
Host: www.googleapis.com
Content-Type: application/x-www-form-urlencoded
client_id=**<your_client_id>**&
client_secret=**<your_client_secret>**&
refresh_token=**<refresh_token>**&
grant_type=refresh_token
It will return the access_token and expires_in
Source
Now the question is how and when to do the http request. So for that I have two ways.
1st one
Store the emailid,refresh token, access token and the current time+3600 seconds. In your database you can schedule to check the expire time in every 5 min and if current time of any user is going to reach(before 5 or 10 min) the expire time, get the refresh token and update the value for that particular user. And during api call just fetch the access token of the user.
2nd one
When user do login in your site, get the access token and current time+ 3600secs and store it in browser cookies. Now before doing any api call just check whether the current time(time when api call is done) is less than the expire time(stored in cookie). If its true then you can use the previous access token else get a new one and again update the cookie. Also you have to put another condition that if the cookie is not present at all then also you have to get new refresh token.