Ability Scores
Operating on the assumption that you'll always be in combat using your Wild Shape feature, you can dump the three physical stats, hitting the minimum 13 for multi-classing as a Monk. I'll be using Half-Elf:
Strength: 8
Dexterity: 13(+1 from Racial Bonus)
Constitution: 8
Intelligence: 14
Wisdom: 16(+1 from Racial Bonus)
Charisma: 16(+2 from Racial Bonus)
At minimum you'll be awarded 4 Attribute Score Increases(ASI) over the course of the game, which should probably be attributed to Wisdom first. After that, assign feats or ASI as suits your interests.
Break Points
I did some of the calculations for average DPR, not accounting for to-hit. Assuming you house rule the natural weapons as unarmed strikes, the break points I've found are:
- Druid 2 (Wild Shape: Dire Wolf)
- Druid 6 (Wild Shape: Giant Elk)
- Druid 9 (Wild Shape: Ankylosaurus)
- Druid 15 (Wild Shape: Triceratops)
- Monk 2 (Ki)
- Monk 5 (Extra Attack)
Additionally add the following breakpoints without the aforementioned house rule:
- Monk 11 (Unarmed Attack D8)
- Monk 17 (Unarmed Attack D10)
Dire Wolf
Druid Level 2
Bite: 2d6 + 3(10)
Monk L2: 1d4 + 3 * 2(11)
Total Average: (Bite: 10) + (Flurry of Blows: 11) = 21
UA NW(Unarmed Natural Weapon) Total: 30
Monk L5: 1d6 + 3 * 2 + (Extra Attack: 2d6 + 3)
Total Average: 10 + 13 + 10 = 30
UA NW: 40
Monk L11: 1d8 + 3 * 2 +(2d6 + 3)
Total Average: 10 + 15 + 10 = 35
UA NW: 40
Monk L17: 1d10 + 3 * 2 +(2d6 + 3) + Quivering Palm
Total Average: 10 + 17 + 10 = 37 AND Quivering Palm Instant Kill
UA NW: 40 + Quivering Palm
(Misses 1 ASI/feat)
Giant Elk
Druid Level 6
Ram: 2d6 + 4 OR
Hooves: 4d8 + 4 (Open Hand Technique combo*)
Monk L2: 1d4 + 4 * 2
Total Average: (Ram: 11) + (Flurry of Blows: 13) 24 OR Hooves(22)=35
UA NW: 35 OR 66(If not already prone: 55)
Monk L5: 1d6 + 4 * 2 + (2d6 + 4)
Total Average: 11 + 13 + 11 = 35 OR Hooves(22)=57
UA NW: 44 OR 88(77)
Monk L11: 1d8 + 4 * 2 + (2d6 + 4)
Total Average: 11 + 17 + 11 = 39 OR Hooves(22)=61
UA NW: 44 OR 88(77)
*Assuming your first strike hits, you can spend a ki-point to knock your target flat, or using your multi-attack, first strike with the Charge feature of the Giant Elk, then Hooves.
Giant Constrictor Snake
Druid Level 6
2d8 + 4 (grappled)
Monk L2: 1d4 + 4 * 2
Total Average: (13) + (Flurry of Blows: 13) = 26
UA NW: 39
Monk L5: 1d6 + 4 * 2 + (2d8 + 4)
Total Average: 13 + 13 + 13 = 39
UA NW: 52
Monk L11: 1d8 + 4 * 2 + (2d8 + 4)
Total Average: 13 + 17 + 13 = 44
UA NW: 52
Ideally go Monk12/Druid8 to max ASI/Feats.
Ankylosaurus
Druid Level 9
Tail: 4d6 + 4
Monk L2: 1d4 + 4 * 2
Total Average: (Tail: 18) + (Flurry of Blows: 13) = 31
UA NW: 54
Monk L5: 1d6 + 4 * 2 + (4d6 + 4)
Total Average: 18 + 13 + 18 = 49
UA NW: 72
Monk L11: 1d8 + 4 * 2 + (4d6 + 4)
Total Average: 18 + 17 + 18 = 54
UA NW: 72
(Misses 1 ASI/feat)
Triceratops
Druid Level 15
Gore: 4d8 + 6
Monk L2: 1d4 + 6 * 2
Total Average: (24) + (Flurry of Blows: 17) = 41
UA NW: 72
Monk L5: 1d6 + 6 * 2 + (2d8 + 4)
Total Average: 24 + 17 + 24 = 65
UA NW: 96
If Stomp instead of Flurry of Blows:
Stomp: 3d10 + 6
Monk L2:
Total Average: (24) + (Stomp: 22.5) = 46.5
Monk L5:
Total Average: 24 + 22.5 + 24 = 60.5
(Misses 1 ASI/feat)

vs

Given the numbers above, the most impressive DPR builds are Druid8/Monk12, and Druid15/Monk5, with the latter being slightly stronger.
However, this determines the average damage per round, which, while broadly useful, stems from a very limited resource of ki-points. Accounting for this, we see the average damage per round over time:

Here we see that, over time, certain builds maintain their DPR much better. Assuming you're averaging 6 rounds of combat each encounter, and 6 encounters per day, with one short rest, the Druid15/Monk5 pulls slightly ahead. This build works at full efficacy for ~1 encounter per short rest, with a reasonable performance on all other encounters. The alternative build of Druid8/Monk12 performs at full efficacy for ~2 encounters per short rest, so your best bet is to choose the build that best reflects the number of encounters you expect to have throughout a given day.
Conclusion
House Ruling the Natural Weapon as an Unarmed Strike still performs lower than the Sharpshooter Fighter builds available, while also being in melee range. You also give up one of the best capstones in the game, for a moderate increase in burst damage.
The clear front-runners for this multiclass are Druid8/Monk12 and Druid15/Monk5, for the priors access to one additional ASI, and the latter for its access to the Monk's Extra Attack feature. Realistically, if you plan to level this character up from 1, you'll probably be most successful with the Druid8/Monk12 as the other build doesn't come online until very late in your character's path.
Given that the levels of Druid are mostly front-loaded skills, you're most likely best off to take at least your first 2 levels in that, and assessing your needs as you level.
Deciding on which of these builds to choose largely depends on a number of factors:
- The number of encounters expected per day
- The number of short rests expected per day
- The likelihood of your party knocking someone prone
- The utility of druid spell slots
- The utility of Ki points (Stunning Strike and Open Hand Technique)
- How highly you evaluate the potential AC improvement with Unarmored Defense
Edit
As a point of reference, a Fighter 20 Battle Master using the Great Weapon Fighting feat averages 93.33 DPR exhausting no resources, while also having great utility, and impressive burst potential that replenish on the same short rest as the builds above. Druid 20 as a Mammoth has a DPR of 25 using Gore, and Monk 20 has a DPR of 22, or 66 within the first 20 rounds of combat a day.
Level 20 Fighter GWF Battle Master
4 * ((2d6(reroll 1/2) + 5) + 10)
4* ((8.33 + 5) + 10)
4* (23.33)
DPR: 93.33
First Round
93.33 + Second Wind(23.33) + Superiority(6.5)*4
142.66
Second Round
116.66