Yes, you can.
(Standard Disclaimer: work with your GM for how that influences game play).
There are no mechanical requirements, but you can make it part of your character's state of being. The following provides some points of departure for your discussion with your DM on how to bring this into your campaign.
This answer addresses "overweight/obese" as asked in the question's text. The distinction between "overweight" and "obese" is worth noting, but isn't necessary for a game usable approximation. As overweight is a subjective description, this addresses "obese" more than "overweight."
Step 1: Body Mass Index (BMI)
From the heights and weights listed for the race you choose, pick the least height and the most weight from the ranges, and then add 10-20% to the weight. This gives you a starting point to make the character "obese" using body mass type that compares height and weight.
Example from the NIH is a 5'8" male weighing 210 pounds has a BMI of 31.9, with 30 being "obese" by current definition.
Note: the BMI model has received plenty of criticism among health and fitness advocates, but for your purposes as numerical model to work from for this game it should suffice.
Step 2: Adjust Ability Scores Limitation to Reflect "Obese"
With the above in mind, limit Strength to average or a bit lower than average (range 8-12).
Some short / stocky / dense / large boned people have high muscle mass which measure as "obese" via the simple tape measure tool NIH provides (they did when I was in the Navy) but they are actually in good shape! (The Navy finally hit BF% about five years after finding this to be such a problem!) A typical example is a short, heavily muscled, and very powerful weight lifter or body builder. A lower strength for your character reflects lower muscle mass / higher fat mass.
Limit Constitution score.
Obesity often brings with it health problems. (a score of 8-12 should suffice).
Step 3: Mechanics.
(This will take some work with your DM).
Being obese doesn't necessarily limit you. I've met some "fat" people who are quite agile. (Example: actor John Belushi was pretty round, but could still do back handsprings during the filming of the movie Blues Brothers).
Do you want to penalize your player with lower movement speeds? You can address a 5' or 10' speed reduction with your DM, or not.
Your agility based skills likely aren't influenced by being obese. But dodging or moving quickly and quietly? Per suggestion from Andras, are you better or worse at hiding ... Stealth checks? Discuss with DM.
From personal experience in playing golf: a few years ago my BMI was 30.2 and my body fat at 30%. (Have since dropped 30 pounds, feel much better, BF % 19-20). My play with wedges, chipping, out of the bunkers, and putting did not degrade. The full swing suffered.
From personal experience (regular life): a good friend years ago was overweight but had amazing manual dexterity. He had fine motor skills to die for, as witnessed by his detail work in painting 25mm lead figures for our table top campaigns. His guitar and piano playing was masterful.
Some "fat" people have compelling, even powerful, personalities. You can just play it that way, or get a Charisma boost in trade for a movement penalty.
This is between you and the DM.
d. Exhaustion. (Per GMNoob's suggestion, Appendix A Basic Rules).
You can apply the effects of Exhaustion (a condition) or make the character more susceptible to it.
Exhaustion (Basic, p. 106)
Exhaustion is measured in six levels.
1 Disadvantage on ability checks
2 Speed halved
3 Disadvantage on attack rolls and saving throws
4 Hit point maximum halved
5 Speed reduced to 0
Warning! Rolling ability checks with disadvantage is a significant penalty during play. This might be overkill.
e. Ability and Skill checks.
There isn't sufficient room to address all of these. You'll need to look at the skills you want and propose some adjustments, or not, based on whether "obese" influences this.
- Lore check? Doubtful.
- Athletics checks? Likely, but your ability scores probably account for that.
- Persuasion checks?
- Intimidation checks?
- Stealth checks?
- Acrobatics checks?
Work with your DM for a good fit.
Step 4: (Humorous)
Make the character a cleric. Friar Tuck would approve.