How can obesity affect my chances at a successful relationship?

Many reasons. Aside from the physical appearance, obesity can adversely affect your health by increase your risk of developing cardiovascular disease, diabetes, stroke, metabolic disease, cancer...It can also affect your mood, energy level. If you are obese during pregnancy, your unborn child has a higher risk of obesity as well...
Relationship. No. Although we live in a a superficial society where thin is considered attractive, it by no means dictates whether or not you will find a great person to be your soulmate.