Wishes: "Be careful of what you wish for, since you are sure to get it." (old adage)
Are there any rules about having dragons as PCs?*
You are in the area of rulings more than rules.
If so, where?
Some places to get ideas for your ruling would be in CH 4, DMG, creating NPC's and the optional villain classes. This is to compare the powers of a dragon with the powers of a given level of NPC. Also take a look at the DMG page 285-289 on variant class options. Then look at how CR is arrived at in the DMG for a given monster that you custom make. That may not give you a balanced new PC type, but it can help you get in the ballpark.
Consider the challenge rating of the dragon. A party of 4 should be able to handle a CR equal to their level. If your players are level 10, adding a young gold dragon is similar to adding 2, 3, or 4 tenth level players to their party. (Depends on how well your players handle monsters in that CR). Adding an adult gold dragon is like adding 2, 3, or 4 level 17 characters to the party, again, depending on how good they are at harder or deadly challenges. See page 9 discussion on Challenge in the Monster Manual.
For example, a party of 4 3rd level characters should find a monster
with challenge rating of 3 to be a worthy level of challenge, but not a deadly one.
If not, how would you go about making it somewhat balanced?
Without a lot of play testing, hard to say. With that caveat in mind ...
You set him a high bar, and he cleared it with a dice roll; D&D does that sometimes. Some points to go with the excellent reply by Cirdec.
How old is he in dragon form?
Proportional or direct age?
- From the table on page 86, if you choose a 1:1 ratio, he is a young gold dragon (age 20, 30, 40, etc). CR = 10. The powers for that are in the Monster Manual on page 115. That's a fairly powerful creature, with flying, and water breathing, etc. A high level sorcerer is no slouch himself.
- If you choose the proportional level of maturity to make him an adult gold dragon, he has legendary actions and can change shape into a beast with CR less than 17 (CR of that age gold dragon). Compare that to a NPC human Mage CR 6 at 11th level, access to fifth level spells. (MM, p. 347). Archmage NPC, access to 9th level spells, CR 12. (MM, page. 342). That is likely not balanced with his fellow adventurers and their abilities.
How old is the sorcerer?
Early 20's? 30's? 40's? 50's? Suggest that he is that age, and thus a young golden dragon. (Depending on the character level of the sorcerer and the party, this may suffice. Or not).
How old of a dragon are you willing to make him?
Humans live less than a century (so 80-100 would be a long life) years (PHB. p. 31) Dragons can live 800+ (DMG p 86). That's roughly 10 to 1 in proportion.
What level is he; what level are the other party members?
If they are at level 13, it's one thing, if they are at 19 or 20 they are in the epic tier. Consider their CR rating using the tools in DMG, p 273 - 279 to compare theirs to his in his new form.
If you allow proportional age (adult gold dragon) and the party wish to keep playing, you may need to scale his abilities into somewhere young and adult. His change shape power may need a limitation, or removal. (Depending on level, allowing him to use that to turn into his old sorcerer self is likely a power downgrade). He isn't a sorcerer anymore, he's a gold dragon ... who can change shape into humanoid or beast form. (At no higher CR than the gold dragon is, p. 114, MM, which is CR 17. That's powerful).
With a bit more work, you might be able to find something in between adult and young, or between wyrmling and young if you take Cirdec's conservative approach. Tweak it by applying various DC's (with a chance of failure) on his Young or Adult powers to keep him within bounds of your other players. Trying to tie this to "PC levels" in the "Gold Dragon Class" is something that would need play testing to get into the ballpark. Allowing him to progress in level to Adult gold looks hard for balance unless you vary skills and attributes. The other players, human, might be dead by then, although elves, dwarves, and half elves may live long enough to see their friend grow into an awesome beast.
If you proceed with this, he's a young dragon with as many dragon years as human years.
The adventures that this change can bring to the party, in terms of how NPC's, lords, wizards, thieves, sellswords, kindoms, etc react to a gold dragon running about could be great fun. He could end up as a wanted man/dragon, his likeness up in all of the post offices on the King's road ...
Suggest also: Disadvantage on persuasion checks for anyone owning livestock. :)
Afterthought: if what you were looking for is a way to have levels of progression for the PC along with the rest of the party, a way to moderate the aspect of power that is HP is to set the hit dice (young) at the sweet spot of young (about 50-70 years old) and then reduce some proportionally for the age the sorcerer is when he turns into one. (If he's 25, Reduce by a fraction so that there is room to grow).
This is if the "PC dragon" is to be able to grow and achieve more ability as the other players in the party progress. Do the same with reducing the number of HD for breath weapons. Or make the recharge harder to score on a die.
If you don't want to do that, then the points Cerdic made about making a wyrmling or a young dragon still give you room to vary the hit dice if you think one is too weak or the other too strong, compared to the rest of your party.