In the last quest, you learn from Kellogg's memories and by talking with Doctor Amari that your best chance to get into The Institute is to find the only person to successfully escape; the scientist Virgil. However, he's camped out in The Glowing Sea, a heavily irradiated area in the bottom south-west of the map. Before even daring to start this mission, make sure you have huge amounts of Rad-X and RadAway. You'll also need some pretty strong Rad-resistant gear so the Hazmat Suit or a suit of Power Armor are essential. If you're out in the open without protection you'll be killed pretty quickly by the rads, so be prepared. If you're taking a companion with you, you'll need one who is Rad-resistant. So, essentially, any companion who isn't a human. Or Dogmeat.
Looking at the map, you need to be heading towards the bottom left corner, or directly south west. The safest bet, especially if you're not using Power Armor or you don't have many Fusion Cores is to find the Electrical Hobbyist's Club which is just on the edge of the Glowing Sea. This will provide a quick way to get back into the area if you have to fast travel out for whatever reason. The best bet is to follow the way point marker on your HUD to keep you heading in the right direction.
By following the way point marker and keeping in a relatively South West trajectory you'll happen across a couple of buildings you can enter. One is a sunken Red Rocket station that contains a Master Safe. The next is the Capsized Factory, which has a few goodies in but also happens to contain several Feral Ghouls who can easily overwhelm you. Throughout the journey you'll also happen across Radscorpions, Bloodbugs and Bloatflies.
Keep following the waypoint marker and you'll end up at Crater of Atom, an impact point from the bombs that were dropped 200 years ago. The people here are heavily irradiated and they are generally not that accepting of you. Find and speak to Mother Isolde, the leader of the Atom cult. Lowering your weapon and speaking with her will result in her pointing you to a cave. The cave is southwest from the Crater of Atom, beyond the edge of your map.
When you get close to the cave, a Deathclaw will attack you. Be careful and be mindful of taking too much damage if you're using power armor. Once the Deathclaw is down, head into the cave.
You will encounter Virgil inside the cave. In exchange for his help, he asks you to fetch a serum for him from inside The Institute. While helping him is not integral to the main story, it will initiate a misc quest called Curing Virgil. You can also opt to kill Virgil or steal from him to obtain Virgil's Rifle. After speaking to him, the Hunter/Hunted quest will trigger.
- JudasBlitzkrieg - The Glowing Sea Walkthrough