You can do it a number of ways:

1. Virtual table top(VTT): Software that allows the players to act as if they are sitting around a table. There are lots of versions of this, I've used maptools to do this, we didnt play online so much as use maptools in place of a play mat, dice and miniatures so I'll talk about it from that perspective. I was the DM and setup the maptools server and clients for my group to play on at my house using my computers as a test. It went well but it's a different experience than playing pen and paper. I can say it takes a hell of a lot more prep to DM using a virtual table top than P&P. Drawing out an 30x30 room for an encounter on a flip mat versus photoshopping a 2d image of a dungeon complete with textures, objects, etc. Is a lot less work, but I'm a bit of a perfectionist when I DM so you could probably cut that down depending on how exact you want the VTT to look. The VTT does the following: displays a map of the dungeon, shows icons for the monsters and players, has chat boxes for communicating, and keeps track of dice rolls.

You can make this really complicated, or really simple. I spent about a month prepping for the adventure, I ran three levels of module B4, The Lost City. I textured the maps, put in furniture, light sources, pre configured encounters including combat detail for the creatures. Each individual icon on the map for each creature I could mouse over and see his light source and vision range, and combat stats, morale, movement.... When I'd move the monsters unit icon they'd have to follow movement rules, their light sources moved with them, if they rolled a to hit, it would add appropriate bonuses. Can you tell I wasnt playing CF that month? I tried to fire up maptools to share some screenshots of how my adventure looked, but it isnt running on my work PC at the moment. Best I can do without spoiling anything is show what the entry level basic map looks like, versus what the DM map looks like after I textured and added detail. Keep in mind, this wouldnt be visible in this way to players, the walls block their light source, so they would see only as far as their PC's vision allows.

DM Map included with the adventure:
[www.dropbox.com]

DM map as I edited it, some of the content such as the traps in blue do not appear until triggered.:
[www.dropbox.com]


You can also just use the map as it comes in the module, then slap down four "H"'s on the map and call them hobgoblins, do all the rest of that on paper outside the VTT. Have the players hit "Roll d20" then add their to hit, tell them if they hit, then say roll a d8 for damage. You can also turn off "fog of war" so light sources and vision range do not matter, etc. Scan in the basic map included with the adventure, and not set up movement rules, etc. It's really about how much time you want to put into it as the DM.

2. One of the guys from my pen and paper Pathfinder group is DMing 5th edition using google hangouts. All players connect with video chat, the DM has two cameras one on himself and one on his flip mat. They roleplay their characters, I think they may use something online to generate random numbers to keep everyone honest on dice.