The first 10 miles of Alexandria's 220-mile cable television system were laid last week, as officials of Alexandria Cablevision Co. predicted 6,000 city residents would watch the system when it debuts in early October.

Sue Cieslak, public relations director for the company, said residents who live in the Fort Ward area, the West End and Seminary Valley in the western part of the city would get the system first because the area's height provides the best reception. The system's electronic equipment is being installed in the West End at the Fort Ward Towers apartment building on North Van Dorn Street.

The $6.75 million system is expected to serve 45,000 households by the end of 1981, and eventually to extend to 55,000 households.

When completed, the Alexandria system will offer 35 channels in a "tiered" structure allowing subscribers to choose how much cable service they want. Service for the basic 35 channels will cost $9 a month. Those channels, plus HBO or Showtime movies will cost $18.95 a month. The 35 channels and both movie channels will cost $24.95 a month.

The company is planning to waive the installation fee of $19.95 for the first few months to lure subscribers. The system will then stretch out along utility and telephone lines, above and below the ground, to the eastern sections of the city.

Cable company officials are hopeful that 50 miles of the system will be operational by the end of October, allowing 15,000 residential units to hook up.