Grandpa is Dash's grandfather. Dash lives in his house. Grandpa had a dream where he was attacked by the Lich's Doom spell. He told Dash of the dream, and gave him the Training Sword. In Story Mode, Grandpa leaves notes for Dash at all of the campsites. He has knowledge of all of the Zones, and the monsters in them. He also seems to have some previous experience with the all of the bosses and their tactics.
He never mentions his son or daughter, whichever one might have been Dash's parent.
At the start of each zone, Grandpa has a tent and fire set up where he has left notes. The start of the first zone is the house instead of a tent, and is a welcome message. It's the only message that doesn't send "-Love, Grandpa" and appears more from the developers.
Dark Forest: "Keep your wits about you! Dark elves and thieves make Dark Forest their home. The skies are infested with bats, and giant spiders build nests here! -Love, Grandpa"
Gloom Cavern: "Spiders, scorpions, thieves and dark elves all lurk in the tunnels and crevices of Gloom Caverns. Some species of oozes are only vulnerable to certain types of attacks. Be sure to vary your approach! -Love, Grandpa"
Deadly Marsh: "Things don't get any easier from here, Dash! Deadly Marsh has earned its name. The Lich's skeleton legion has a strong foothold here and they'll do their best to stop you from continuing on. -Love, Grandpa"
Castle: "This is your most difficult challenge yet! You'll confront the Lich's most powerful soldiers: skeletons, ogres and dragons. There's an army of deadly dark knights, and you'll have to fight their commander, the Demon Knight. -Love, Grandpa"
Dark Castle: "This is it, Dash! The Lich is going to throw everything he's got at you in order to stop you, so be sure you're at full strength before you proceed. You can restore peace to the kingdom, my grandson. Fight bravely! -Love, Grandpa"
One more message appears after clearing the final story mode level. It references some meta-things from the game but still ends "-Love, Grandpa":
Dash, you've done it! You've defeated the Lich, but there's more to do! Try for a top score in Mini-games and Challenges. Try the Classic Arena, where the challenge never ends! And look for new areas on the map in future updates. Have you gotten 100% on the World Map yet? -Love, Grandpa"
- The Lich keeps coming back because of a phylactery
In his final note, Grandpa tells Dash he has defeated the Lich. But he can still be fought, in classic and story modes. Grandpa also knows all the enemy and boss tactics. The Lich recognizes the Memento item, which says it was "a gift from a long lost loved one". Based on this, it seems as though Grandpa's child/Dash's parents, once fought all of the enemies and returned home victorious to be able to tell Grandpa the tale.
"But the Lich just has a phylactery and came back to life!", I hear you scream! That's possible; the item given to you for completing the world map is the explorer's compass. The world map points north as the top of the screen, and this game never rotates, so north is always "up", however the explorer's compass points to the northwest, so maybe it's a key to finding the phylactery. Or maybe the Memento is the phylactery, as the Lich yells out "Noooo! Where did you find that ...! Arrghhhhh!" as if it's a painful/important item to him.
2. Everything happens over and over again because of a time loop
However! When Dash dies, he can be "revived" - many items and abilities use this term. When he dies for good, he just returns back to the main screen, ready to go again anyways. No matter death or Victory, the Lich always appears again and Dash is always ready to fight again. The final key here is Becoming Legend. When you choose to do so, you are told you'll join the "elite dashers". This implies there are other "Dash"'s in the universe. Some of which may be appearing when Dash actually dies and returns to the main menu, others that may replace the one that becomes a "Legend": a story.
Why would the Elite Dashers choose to end their lives and let a new Dash replace them, when they know they could live forever? At some point of infinity, the spirit would lose the will to go on, and by becoming Legend they hope to inspire the new Dash with their story (and improved ability that can't be attained any other way) to continue the fight, in the hopes that one day he might escape the time loop.
3. Grandpa knows about the real world
Based on his final message, Grandpa appears to know much more about the world than a regular character inside the world. Because he references these things casually, he has likely told Dash about them before.