For a system using a single Radeon HD 6770 graphics card AMD specifies a minimum of a 450 Watt or greater power supply. The power supply should also have a combined +12 Volt continuous current rating of 24 Amps or greater and have at least one 6-pin PCI Express supplementary power connector.
Total Power Supply Wattage is NOT the crucial factor in power supply selection!!! Total Combined Continuous Power/Current Available on the +12V Rail(s) rated at 45°C - 50°C ambient temperature, is the most critical factor.
You may find power supplies on the market that supply more than enough Wattage to run the system. However, many of them lack the Amperage capacity on the critical +12 Volt rail, which is necessary to properly power the critical components in the system (i.e. CPU and GPUs). This is the reason why graphics card manufacturers overstate the power supply wattage, usually by at least 50 Watts, to take into account those power supplies that have the weaker +12 Volt rail(s).
Your Dell Inspiron 537 Desktop's 300 Watt power supply, with its combined +12 Volt continuous current rating of 18 Amps and with no PCI Express supplementary power connectors, is insufficient to power your system configuration with a single Radeon HD 6770.
You'll have to increase your budget and purchase a new power supply unit.
The following power supply meets the requirements :