If you write this same question on Google You will get a long list of tools, applications and browser addons for running a spelling and grammar check and a link test for websites.
Personally, what I do is copy the content of web page and paste it in a Word Processor like MS Word or Libre Office Word and run the spelling and grammar check in it for finding the issues. For Final verification I go through the content to make sure it is correct.
Sometimes the spelling and grammar are correct but the sentence is wrong or is unfit. For this part I don't rely on tools but my knowledge of English.
For link testing there are countless tools of which I prefer the following,