Bleeding gums are most commonly caused by plaque (food debris and bacterial colony) around teeth. Plaque causes infection of the gums which makes it prone to bleeding. I will advise you to visit a dentist who will do professional cleaning of your teeth (scaling) and may also prescribe you some medications. Thereafter try to maintain good oral hygiene by regular brushing & rinsing. The problem of sensitivity is also likely to get cured after this treatment. Hope this helps you.

Please consult Periodontist and get your gums conditions evaluated... It might be Gingivitis or Periodontitis problem.