You can not see the Northern Lights at the same time as the midnight sun.
You can only see the midnight sun during summertime in Iceland, that is in May, June and July. (Technically only in June, but it’s bright at midnight in May and July). August also has bright nights, but it gets dark around midnight in August. In Iceland the hours of daylight are always getting shorter or longer by a couple of minutes each day. The shortest day of the year is around the 21st of December and the longest day is around the 21st of June.
Depending on if you are in Reykjavík or in north Iceland, such as in Akureyri, there’s a difference of how long the days are by a few minutes. You can experience the midnight sun wherever you are in the country, but the further north you go in the country the days will be a little brighter in summer and a little darker in winter (not so much that you’ll notice a difference though).
When the midnight sun occurs, the nights don’t get any darkness in Iceland. In order to see the Northern Lights the sky needs to be dark. The best time to see the Northern Lights is anytime the nights are dark in Iceland (so from late August until end of April). The best time for the midnight sun is at the summer solstice on June 21st.
Read more about the Midnight Sun here and about the Northern Lights here.