I am a Level 3 Fighter Battle Master in a 5e group. When I leveled up I became proficient in Smith's Tools and have since got an apprenticeship with an NPC blacksmith to hone my skills, which I do during my downtime. Is there some way to calculate how my skills in smithing progress over the course of my apprenticeship? My DM has given me sole responsibility of doing this.
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The PHB p.187 gives rules for training during downtime.
By the book proficiency in 5e is binary with you either getting your whole proficiency bonus to the check or none at all. That said it would be a trivial house-rule to instead say that as you train you slowly become more proficient and have this represented by slowly getting to use more of your proficiency bonus on checks.
The objective answer: in 5e, skills don't progress in that way.
That being said, you're already well into house rule territory, so I am unable to provide direct guidance. Here are my thoughts on the question, and a possible solution.
Skills in 5e don't progress. You either have a skill or tool proficiency, or you don't. The primary way of progressing in any proficiency is simply by leveling. As you level up your proficiency bonus increases, reflecting your natural character progression and improvement in all proficient areas. Typically adding a new proficiency through training takes a long time as Ceribia's answer states.
However, if you want to play this out in some way, you will effectively need to change how tool proficiencies work. D&D 5e doesn't have a lot in the way of downtime and crafting rules, presumably because the designers wanted to avoid encouraging "merchant simulator" type games. There are other systems out there for that type of play.
D&D 5e is a heroic adventuring game, and there's not much heroic about standing and sweating at a forge all day. Since a novice or journeyman blacksmith spends most, if not all, of her time at a forge working towards expert and master levels, she wouldn't have much time to devote to adventuring. Likewise, unless your character has months or years in between adventures to devote to spending 12 hours a day at a forge, she likely won't progress very much at all in her blacksmithing skills.
That being said... your character will get better at blacksmithing by leveling up. I know it seems counter-intuitive, but since your proficiency bonus in tools use comes from your character level proficiency bonus, your character will become a better blacksmith simply by adventuring and gaining experience.
Proficiencies progress in three ways:
1. Having the proficiency. As others have pointed out, proficiency in 5e is a binary thing. Except it's not. It's explicitly quaternary, with 0, 1/2, 1, and 2 being the available digits. That is: you can be not-proficient, you can be half-proficient (Jack of all Trades, PHB p.54; Remarkable Athlete, PHB p.72), you can be proficient, or you can have expertise (Bard at PHB p.54, Rogue at PHB p.96).
You've already hit the third stage here, so your remaining (existing) way to improve as a smith is to develop expertise. I think you can make a strong case to your DM that 250 downtime days and gp could net you expertise on top of your proficiency.
2. Improving the relevant stat. This doesn't apply to a tool, so doesn't help you. But if you'd taken an apprenticeship as a natural healer, for instance, and hoped to improve your Medicine (Wisdom) skill, it could have come into play.
3. Improving your proficiency bonus. This happens when you level up, and only rarely. So can you level up faster? If smithing is the sort of thing you like to do and your character spends much time on you should talk to your DM about whether your smithing exertions should be gaining you XP. There are plenty of suggestions in the DMG (pp.260-261) about non-encounter (not just non-combat) XP awards, and this might be one of them.
Just beware: if your downtime activities earn you XP you may set off a "gold rush" at your table for downtime->XP-converting activities. When all you wanted to do was relax a bit at the forge.
Maybe you're just a better blacksmith...
Right now you are crafting at a rate of 5gp per day. As you pour more and more time into this profession you could propose to your DM that your work is increasing in quality/you're increasing in renown, so your proceeds should be increasing. Given that "Crafting Magical Items" allows one to craft at a rate of 25gp per day, I'd say there's some room for your rates to reflect some amount of better-than-average smithing without stretching things too far.
If you're looking for slightly more detail, there's "Running a Business" at DMG p.129. Applying a positive modifier based on your proficiency bonus to the table found there would nicely simulate increasing one's professional skill.
...or maybe you're an awesome blacksmith.
This one's the nuclear option: create the Blacksmith class and take a few levels. Resources on how one might do this can be found in various Unearthed Arcana articles, specifically "Class Design Variants", the design notes for "Ranger", and "Prestige Classes".
In this case you've got the ability to add amazingness to your blacksmith in the form of class features, but it comes at the expense of your core class features. A fair trade? That's up to you and your GM to design and then decide.