I have a hard time telling what a "Budget" deck is these days, and the price for some cards is fluctuating a lot recently (Fiend Artisan and Underrealm Lich). So if this is too expensive for budget please let me know
Concept is to mill yourself to throw your good cards into the graveyard, and bring them back to your hand or the battlefield. Since the deck is mostly creatures it can also combo off with Syr Konrad, the Grim on the self mill and pulling cards from graveyard, but can also work with general trades and some sacrifice. It goes without saying that cards like Leyline of the Void and similar will hurt the deck a lot. Hushbringer can be problematic due to on death and etb mill triggers. Although the deck doesn't have as much casting from the grave or going directly to battlefield, so cards like Grafdigger's Cage are not as problematic.
Had some serious problems with weenie aggro, so I added some low cmc board wipes in sideboard to pull in for bo3s. Still looking for a better low cmc board wipe though as Cry of the Carnarium exiles creatures and that can delay my graveyard if I wipe my side too (most of my early drops have 2 or less toughness), and Witch's Vengeance only works against tribal. There is also Find / Finality, but by the time I have the mana for the wipe any real aggro would have won already unless I focus more on ramp.
Noting issues against control planeswalkers too, since my only real kill spell is Status / Statue. Looking at other options to help with that such as Questing Beast, Eat to Extinction or more extreme Casualties of War. Murderous Rider was an option here too for the adventure, but since this specifically avoids the graveyard it seemed like a non synergistic choice.
Shoved in Skull Prophet since it can help with some ramp/mana drought while having the ability to mill myself (potentially triggering Syr Konrad, the Grim) or attack with the 3 power. Moved Foulmire Knight to side deck for now since the extra draw isn't terribly reliable and would prefer graveyard stuffing, but having deathtouch is nice for chump blocking. Could look at Leyline Prowler but the versatility is nice (minor health, deathtouch, mana for slight ramp). I also moved one Kraul Harpooner to side deck, that way I have some in the deck to sac and search or pull form graveyard if needed. Then I can swap out two other cards next game in a bo3 since I've found them useful for clearing high toughness flyers and I can bring them back easily to cycle if they keep dropping more flyers.
Could replace Pulse of Murasa with something else, but I've been wanting to find an excuse to use this card. Being able to pull from the graveyard is a roundabout way for card advantage/draw, if it is a creature it can also trigger Syr Konrad, the Grim, and the chunk of health can help stabilize against early aggro.
Considering adding a few Fabled Passage since I have 4x in arena, but haven't really run into color problems that often due to the dual lands and mostly equal basic land ratio.
Plan to replace Jungle Hollow with scry lands as I get more.
Tempted to find room for a 4th Underrealm Lich
Probably won't put Storrev, Devkarin Lich in. Left in the maybe pile though since it is not horrible for 4 cmc and can bring stuff back, but it doesn't mill me and takes at least a turn to get a card back. It feels like it would fit better in a deck with more buff cards or red for haste, but maybe it turns out to be a good card and I'm just undervaluing the power.