In order to create “Alert me” option you have to go to the ribbon and click on more option and select Alert me”
Following is a current process for “Alert me” two actions required.
Step I
Step II
But I want some think like following:
Step I only
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If you want to create a custom button the users can click to subscribe for specific list, create your HTML button with a link to the following URL:
https://sharepointsite/_layouts/15/SubNew.aspx?List=%ID%7D&Source=%2F_layouts%2F15%2FMySubs%2Easpx
In this URL the ID is the ID of your list. You can rather easily get this URL by accessing to https://sharepointsite/_layouts/15/SubChoos.aspx, selecting the wanted list and clicking Next. The URL on the following page is what you are looking for.
Note that if your list is on a subsite, you should adjust the URLs to include the path to the subsite.