You love your dog but you want a nice back area to enjoy. I'd leave the porch as it is. Think of a solution for the large-dog-back-grass-mud-area that will result in no mud. From here it looks like the dog will still be attracted to the mud around the patio and track mud across the new patio and into the house.
Maybe you can remove all grass and place weed cloth and small gravel all around the new patio? or contract for extra cement & install a fenced dog run where dog can be placed?

Or .... EasyTurf! Full grown dogs are no match for EasyTurf ( http://www.easyturf.com). We are the #1 choice for pet owners because of our look, feel, durability and most importantly for pet owners DRAINAGE! Check out this video:
Mud will not be a problem ... nor will bugs, mowing, fertilizing, weeding ...
Let me know if you have any questions.
Cheers!
George
http://www.houzz.com/projects/82915/Backyard-Installations

The yard is pretty large and I'm worried that turning it in to gravel will be cost prohibitive. I like the idea of expanding the concrete patio with some gravel. Hopefully, this will alleviate some of the problem or at least allow for a few extra mud-free steps before he reaches the porch area.

Some good ideas. Thinking about keeping the porch enclosed but adding larger windows. For the patio area, this is an idea that I am considering. I would add an larger gravel border but the concept is well illustrated.

The gravel border looks nice, but seems like it will it scatter all over the cement esp. with dog.

Hello Eric
This is a cute back room, but it does look like it has been surrendered to your dogs muddy feet.
Where is the water creating the mud coming from? It looks as though water is collecting directly at the foot of the steps. Before having that much concrete installed I would check on what will be happening with drainage when the patio is poured. Simply making certain that the water is allowed to move downhill and not collect in inappropriate spots may be your most cost effective, long term and least maintenance solution.
Turning large areas of the backyard into gravel or crushed stone will simply give you a rash of other problems, from weed seeds carried into and germinating in the gravel to the additional heat the stone will collect, making the back yard temps hotter.
Libradesigneye has some interesting ideas for adding interest and value to the concrete.
Think about where the dog will "fit in" to your patio scheme - where the dog will have his water bowl, his couch, his run, etc. There might be a space adjacent to the patio defined as "this is where the dog gets cleaned off." This would be where gravel would be best used.
Whether you turn the back room into a screened porch or enlarge the windows looking into the yard your thought to make changes that will give the room more purpose, more of a feeling of comfort are good. Modifying and furnishing it to make it more definitively a space for people to enjoy, or a part of the "back yard," and not just a through-way for the dog will add a lot of value to the house and add exponentially to your enjoyment of it.
Have fun!

I think i would turn the porch into a dog wash and i'd go sit somewhere else.

I had a small entry to my back door like you and with my 2 dogs was always muddy. I made a low deck across the entire back of the house. The dust and the mud has cut way down and I have a place to wipe them off before they come in. Still not perfect but it has helped plus I have a place to hang out which I've really come to enjoy. I would love to have a doggie bath someday. The pooches were nice enough to pose for me this morning. Hope this helps.

Enclosing the back porch and having a doggie door would make it possible for them to get inside out of bad weather if you aren't home.

If you could fit in a dog wash station, you'd love it. I've worked with the animal shelters for years and they use industrial sized fixtures and indusrial deep sinks. For large dogs, there is the walk in shower type. https://www.houzz.com/photos/dog-shower-phbr0lbl-bl~l_39247

Doggie door leading directly to shower station for the big guys.
https://www.houzz.com/photos/dog-shower-phbr0lbl-bl~l_39247?fi=8

Project update:
1) We opted to regrade to ensure that water is flowing away from the house.
2) Installed a proper gravel foundation for a dyed concrete patio.
3) Poured a large patio and will use the fill dirt to soften the edges.
4) Picked up some outdoor furniture on the cheap by taking advantage of fall clearance sales.

Total cost (including furniture): About $4,000

Still to early to tell for sure but the mud problem appears to have been completely resolved in the house and on the porch.

A few more photos.

A few more photos.

Regrading was a smart idea. Looks good so far! Wtih plantings, this will be a very nice backyard. Will you have a firepit somewhere?

I like the idea of larger windows for the porch. In the winter, it will be more like a sun room.

Nice! Some Japanese style landscaping touches might look nice.

Wow! That looks beautiful! You did a fanatastic job! It changes your whole yard, and look of house.

The only thing I was thinking is if you are going to have parties....................invite me!! Seriously, now.

At some point it might be a good idea to put in a sliding door that will go into the wall on both sides, and widen the steps. When people are going in and out to the bathroom or with food, you won't be bouncing off one another. Also maybe another entrance of the right side if needed.

If you change the steps, it would also be nice to have some kind of area outide the door, to sit something down before going down the stairs.

Again, your place looks beautiful!!

We just did the same thing! My dogs love to bask in the sun. Yes, the med tracking will decrease. It did for us when we added a patio to the last house. Congrats! Your patio is beautiful! Now, find "pet friendly" plantings. That is where I am.