Highest average roll
Generally speaking, talking about the highest possible roll and the highest average roll are the same thing, you have a bonus (calculated by your stats and proficiency or expertise) and you have a variance associated with the dice you roll (always a d20 in this case). The variance will always mean you get some number between 1-20 (average 10.5) then you add a bonus.
Let's say the tools you use require dexterity, step one is to max out your dexterity on your character. Next you would like to get proficiency or ideally expertise to get the maximum bonus possible.
A note about Artisan's Tools
Most tools do not have a stat bonus officially associated with them, talk to your DM about whether tools use DEX, INT.. etc.
An example build
6th level Artificer
A 6th level artificer gets the following feature:
Starting at 6th level, your proficiency bonus is doubled for any ability check you make that uses your proficiency with a tool.
Let's say that your chosen tools require DEX, get your DEX as high as possible. For this example we can use 18. At level 6 your proficiency bonus is +3 so we would have DEX +(3 x 2) - totalling +10 in this example. With these stats you would have a high roll of 30 and an average of 20.5
When you have advantage, your average goes from 10.5 to 13.825 (reference) before adding a bonus. You can use the Lucky feat to simulate this effectively.
Use Enhance Ability
Enhance Ability Will allow you to gain advantage for any type of ability check reliably, this also stacks with Lucky. It's worth noting if you Lucky a roll with advantage this gives you an average of 15.49 (reference using the string "3d20D2")
Talk to your DM about if they would allow advantage for Artisan's Tools under special circumstances, such as inventive roleplaying or having special equipment (such as working in a laboratory).