To build a career in foreign banking You should pursue MBA Degree (Finance, International Business, Banking) or Diploma in Banking and Finance, Economics (Bachelor's and PG course). Financial certifications like Chartered Financial Analyst (CFA), Financial Risk Management (FRM), Professional Risk Manager (PRM).You can also try connecting to linkedin profile with people working in Foreign bank and stay upto date with job opportunities and other guidance tips. Upload your resume in job portals like naukri,monster,times jobs.
The following foreign banks provide services in India now
Standard Chartered Bank
Royal Bank of Scotland
Bank of Bahrain and Kuwait:
Bank of America