There are many ways you can do it. Atmosphere, descriptions, clothes, situation. all these things contribute to a sense of experiencing the past and all are important to the authenticity of the text.
One way I don't attempt to recreate the past is by language. Or at least not very much.
I've had reviews that complain that much of the vocabulary I use wasn't in use at the time the book is set. Of course it wasn't. If I were to try to use truly authentic language, it would require a vast amount of research - of the reading ancient texts in gothic script variety - and I don't think anyone would read my books. They would be too difficult to access. And I see my role as one of making the past accessible, enjoyable and easy to read about.
In the case of my Viking books, they are 'in translation' anyway, as the characters weren't speaking English, but Norse. And in the case of my Tudor novel, The Lady in the Tower, the language people spoke would have been so dramatically different to our own that we would struggle to understand it. It would be somewhere between Chaucer and Shakespeare.
As far as I'm concerned, language is the medium we use to access the story and not really a part of the authenticity of the setting. Of course, that said, it's still a minefield.
I have to be very careful not to include anachronisms, for example. It's more easily done that one might imagine, because we take our everyday items and the names we have for them so for granted that we scarcely see them. Thus my copy editor found and alerted me to a mention of trousers in my second Viking novel. Norse men wore leggings and tunics. Of course, I know perfectly well that trousers as a clothing form didn't exist until the Victorian age, but that kind of slip is so easy to make if you're cracking a joke or using a catchphrase.
And then there are the words that have changed their meanings. I've had to be really careful with these in my latest novel, which is set in the early Georgian era. In those days, a dress was a gown, a wardrobe was the clothes you owned and the piece furniture you kept them in was a closet. Skirts refered to the section of a man's coat below the waist; women had petticoats or 'coats', not skirts. I knew all these and others and had a list beside me as I wrote. Nonethless, when I did a search of the document once it was finished, I found six instances of the word 'dress' that had evaded my attention.
Getting these words right - the era-specific references - is very important to me. I think it does help to recreate the past. I did a great deal of research on costumes, furniture, food, literature and buildings, as well as lifestyle. Accuracy is valuable. Especially when readers are basing their knowledge of the era on your work. And of course modern slang is completely out of place in a book set in the past.
But if you want to tackle the old authentic language, don't come to modern historical fiction; seek out the plays, poems, stories and novels that were written back in the period you're interested in. Personally I enjoy reading many of them. But it's an altogether different experience.