The Three Pillars of Adventure solo

The gameplay elements of 5th edition D&D are broken down into the "Three Pillars of Adventure". These are:
1. Combat
2. Exploration
3. Social Interaction

To have an enjoyable solitaire gaming experience, it is important to adapt these elements into a solitaire framework

Combat

Combat is the easiest pillar to perform solo. 

 D&D combat contains little to no hidden surprises and the dice system introduces a high degree of randomness, making this a perfect system to perform solo. The player only has to play both sides intelligently, and let the dice fall where they may (or not, you're playing solo, nobody cares if you fudge results). Any "surprise" variables such as monster special attacks or reinforcements can just be introduced through dice mechanics. 

 But D&D is more than just a skirmish level wargame, and if the game was only combat then it would get boring very quickly. To contextualize the combat there is dungeons and exploration.

Exploration

Exploration is possible to play solo with random generation of layouts and terrain and random encounter creation. A simple table of encounters and any number of procedural generators can be used to accomplish this task. But there are hidden elements in dungeon and wilderness exploration that need special consideration to not ruin the fun of surprise.

 Trap placements, secret doors and hidden treasures all exist as knowledge meant only for the DM, to be revealed to the players only through clever play and after taking the proper precautions. It is possible to emulate these elements through solo play, and with a little work it can feel almost as natural as learning about them from a DM.

The AD&D DMG has a great list of traps in Appendix A Table VII that are eminently usable solo, and OD&D Supplement I: Greyhawk devotes an entire chapter to them. The D&D board games use face-down trap tokens as hazards to PCs. Their placement is known but the effect is not, and generally they are placed at the edge of a tile, halting player progress. The player is then faced with a single chance to disarm them, or otherwise pass over them and suffer the damage.  Random event and encounter cards can also simulate the unpredictability of a dungeon crawl. Similar random events and hidden hazards can be adapted for wilderness play, making overland exploration a fun part of solo play as well.

Pregenerated dungeons can be used to play solo effectively, with a bit of work. For this method its ideal to have a Player's version of the dungeon map that only shows the basic layout, and a DM version that has all the traps and secret doors and treasures listed. It really depends on how well the dungeon keys are written, the kind that are best for solo play keep the player information in the top two paragraphs and the DM only or secret information below them. Modules that use a lot of boxed text are great for this method. To play through this method, first move through a dungeon room and make a note of everything you would do as a player - where you check for traps or treasure, what you decide to look for, how you decide to exit the room or listen for monsters. After making all your player decisions, then read the DM information in the room key and flip over to the DM map and note if any characters triggered a trap or found secret objects, and adjudicate victory or loss appropriately. 

Social Interaction

Social interaction is the pillar least adaptable to solitaire play. NPCs generally only do a few player-facing tasks: provide an opportunity for shopping, combat, help or hinder PCs, become followers, and provide rumors or push the story forward.

You can approximate NPC interactions by using reaction rolls and morale checks, but an important part of social interaction is learning rumors, finding out new information and hearing the story of the characters and the game world. All this information is meant to be the sole purview of the DM and can't easily be randomly generated.

  The Mythic Game Master Emulator uses event focus and event meaning tables to generate story content, and a worksheet of scenes and NPCs to determine the outcome of a story. Rory's Story Cubes or other methods of creative writing also fulfill this experience. The 5e DMG devotes an entire chapter to creating story quests and progressing through NPC motivations and plot twists, and fills it with tables for the explicit purpose of random generation. 

 Personally, I don't write out huge dialogue exchanges between PCs and NPCs. I generally determine the disposition of the NPCs, any applicable persuasion attempts by the PCs, and the results. If the PCs fail, then no new rumors are gained or resolved. But if you want to flex your creative writing muscles, this is the place to fill in the dialogue.

Using a premade adventure module can help fill out the story aspect of the game, but adventure modules are written entirely for a DM and generally don't separate the player facing information from DM-only information. Going across the wilderness travel or dungeon exploration part of an adventure module is definitely doable using the above mentioned procedures, but when it comes to the actual story you basically have to read it beforehand and are "on rails" throughout the entire experience. Personally, I find reading an adventure module to be catatonically dull, and I use solitaire procedures to work my way through a module in order to prep it for a live game.

Ironically, the more experience you have playing D&D with a live group, the better your solo play experience will be. D&D, the nerdiest game ever made, requires you to socialize.