If “j’ai l’impression” is followed by an infinitive, use de (or d’ in front of a vowel or mute H):

• J’ai l’impression d’avoir oublié quelque chose → I feel like I forgot something
• Elle a l’impression de perdre son temps → She feels like she’s wasting her time
• Ça donne l’impression de flotter → It makes you feel like you’re floating in mid-air

If “j’ai l’impression” is followed by a conjugated verb, then use que (or qu’):

• J’ai l’impression que ça ne va pas durer → I’m under the impression it won’t last long
• Avez-vous l’impression qu’il se mettra à pleuvoir bientôt ? → Do you think it’s going to be raining before long?
• On avait l’impression que tu ne reviendrais pas avant une semaine → We were under the impression you wouldn’t be back for a week

Concerning adjectives (or other types of words) used with the phrase, they would then define more precisely the type of impression felt. I’d say they would usually preceed the word impression, though.

• J’ai la persistante impression que l’on m’observe → I persistently feel someone is watching me
• J’avais toujours cette impression de d’être sur le point d’un grand changement → I always had this impression of standing on the verge of some great changes

Also take note of the very appropriate comments by aCOSwt below. I shall integrate them into the answer when I have more time.